Sometimes famously so.

I am not aware of any situation where the use of an apostrophe to indicate a plural would be correct in modern English, or in any modern dialect that I am aware of.

You are correct, but, this is an oversimplification. Let me explain. :)

Duality and Infinity
You are probably familiar with the impact of infinity on the concept of probability, namely if something is possible, given an infinite amount of time, it is inevitable.

The problem with absolute arguments is that nothing in an infinite universe is dualistic (i.e., a binary proposition). Cue discussion on quantum physics, or — if you prefer theology — Manichaean duality. (This is where Betrand Russell & Richard North Whitehead, with the 1910 seminal treatise Principia Mathematica, came unstuck. Even in their artificial construct — read, a finite universe with defined laws — like analytic[1] mathematics, it was impossible to create an axiomatic definition to split every member of the universe into either one group or its complement, representing a property (i.e., something an object has by definition: an existential factor).[2] This is what I mean by the unavoidable misapprehension when trying to map the infinite onto the finite; at some point the reduction of information that is necessary for a summary to be useful — chunking — must impact the wealth of data it purports to represent. A simple mean can be enhanced with additional metrics, like the standard deviation, for example; think of the average life span of a Classical human, Socrates was executed in his eighties, and the elite troops that Alexander used to conquer the Earth were in fact inherited from his father; they were not young men, all older than the average life expectancy, which was short, simply because an average includes infant mortality, which was a major cause of death until the twentieth century. :)

Think about what language is. The primal purpose is to mediate thought between individuals. Different languages have different shortcomings when translating thoughts; English has "I love, I am loving, I do love", whereas Latin merely conveys all three with the one mnemic phoneme, amo. (But, since Latin does not rely on word order for semantic transference, a writer is free to use word order to convey extra meaning, so this seeming reduction in precision may be circumvented with care.)

So, bearing that fact in mind, think of what a possessive is.
Does it refer to the properties of all members of a group (hence, obviously plural), or do we identify a single case (of this particular one)? Clearly, it depends. When constructing a compound noun, the rule is to use a singular adverbial, like toothbrush (not teethbrush). This helps to differentiate the compound plural (toothbrushes) from the brushing of many teeth (does teeth-brushing imply one brush or more?)
Was the apocryphal greengrocer referring to a bunch of bananas, or the singular banana, with a price per weight volume? (Both an individual banana, and a bunch, have a mass that can be calculated as a price per pound.)

A noun is a cognitolingual icon with a real referent; and a suffix indicating ownership (the genitive case) is directly comparable with the equivalent adjectival compound phrase, e.g. a driver's licence and a driving licence (in this case the present tense participle[3] gerund, is acting as a descriptor for the noun "licence", which is clearly a noun in those dialects that distinguish between noun and verb by alternating the spelling between /-c-/ and /-s-/, as is universal for the noun-verb couplet advice-advise).

Is one method of conveying the meme better? Should speakers aim to use one method over another? (California raisins, instead of the adjective Californian, and the illiterate California's, demonstrates this process. Though, note, an exception is easy to construct for using the last term, say, if there was a need to refer all the raisins located in California.)

To your point, it is pretty rare to see anyone nowadays use an apostrophe in a numerical representation — the seventies = 1970[']s — though this was standard practice only a generation or so ago. (For example, if you read 1982 Pulitzer Prize winner Douglas Hofstadter (1979), Gödel, Escher, Bach: an Eternal Golden Braid, Mr Hofstadter was probably — unintentionally— one of the leading causes for the rethink: he writes about "A"'s, when a modern typographer would signal this with As; note the removal of both the quotation marks, replaced by the use of italics, and the apostrophe.)

Perhaps looking to how other languages deal with possessive-plural demarcation would be helpful? Latin, for example, has the same synthetic (i.e. the adjustment of a word to indicate its part of speech) construct for both the singular genitive and plural nominative cases (for the first three declensions), so that puellæ means both an object's ownership, "of the girl", and a predicate referring to "multiple girls". In Latin, there is no distinction between the two. Does there need to be? In the specific instance, obviously the speakers & writers are confident in discriminating between the two distinct meanings; the alternative is adding complexity —a different suffix or sentence construction — which will mitigate against simplicity, and impact the language's ability to be learnt and its eloquence. If the reality was different, then (according to Deacon) the language would have adapted differently.

All these questions are not normally verbalized when learning a language. They are implicitly learnt. Nevertheless, they are not unusual encounters for the speaker. At some point (in an infinite universe) the speaker will need to differentiate between two instances of reference that perhaps they language doesn't permit. I usually choose to recast a sentence, rather than use a clunky construct, but spoken language is different.

To my original reply:
The needs to communicate and coöperate drive human evolution. (It is "over-selected", meaning feedback both positive & negative leads to more.) New speakers will change the language. (This is what Terrence Deacon discovered and, to my mind, cogently proved, with his treatise on the coëvolution of the human brain and the symbolic language we all use to communicate. This theory has been around for two decades but, because of its clear refutation of Noam Chomsky, the darling of linguistics, and the politics of science, it has not penetrated into the public domain.)

This is why grammarians cannot dictate how people use language, it is a Sisyphean task that will ultimately fail.

Even with, say, the Queen's English, there are more speakers of English as a foreign language than native speakers, so what Anglophones in Singapore or Bangalore do is less influenced by the diktats of Oxford or even Harvard than the pragmatic requirements of linguistic intercourse. However, it is not impossible to create standards, however temporary, that allow current and future readers the ability to glean an understanding of the writer's intention.

That's why I also maintain that punctuation is a method of assisting the reader. In fact, it might be possible to map the current text-speak lack of punctuation and use of homophone brevity (UR for both your and you are are less explicit than their English counterparts, but perhaps the decryption is something with which future readers will be adept, and so not such a mistake as a modernization?) to an index of amity within the polity (or amongst the members of the Sprachtbund more generally), in direct proportion, so that better punctuation (n.b., not necessarily more) reflects a more civil discourse and autochthonal society.

edit:
typography :)
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[1] Nota bene, analytic in philosophical context, as opposed to the earlier linguistic reference, means a construction whereby truth is establish by definition; scilicet, one defines, say, a human as a mortal being, meaning it is now implicit that humans must die (to be regarded as human, in this context). By definition — or analytically — an immortal human is a contradiction in terms.
[2] The example used is hair cuts: if every one in a village — with only one barber — either cuts their own hair or has the barber cut it: is the barber a member of the barber group, or the self-cutters? Or both? Or neither?
[3] A participle is a verb used as an adjective. A gerund is a verb used as a noun, like singing in the sentence I like singing.

If the property is simple (I think the technical term is something like "primitive recursive"), then one can easily split character sequences into two categories.

For example, if we look at a formal system like arithmetic, we can easily split them into well-formed statements ("1+1=3", for example, is well formed) and not well-formed statements (like if we just threw a bunch of symbols together, like +=1-2= :"). Or we could divide statements into those with existential quantifiers and those without.

I could also point out that the formal systems in question are actually infinite; there are (countably) infinitely many statements in a formal system, and in the case of arithmetic, an infinite number of theorems. (Consider the thoerems 1+1=2, 2+1=3, 3+1=4, etc.; this is an infinite sequence of theorems of arithmetic.)

(Also, I hope you realize that your post is far more complex than it needs to be, and many readers are not going to understand what you wrote as a result.)

I believe that isn't necessarily the case if the event almost never happens (that is, it has probability 0, like the chance of a randomly chosen integer (with no bounds) being an even prime).

TLDR; a language can map a relationship as the noun's possessive case, or an adjectival phrase, for example, and the semantic payload (salva veritate) is equivalent but the choice is often arbitrary.

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Not all problems have easy solutions. I am trying to be both concise & correct, though I err on being both consistent (contradiction-free) and complete.

[Prolegomena ominatio] Trigger Warning: dangerous brain-torsion ahead. Buckle-up!

Both correct and incorrect
Take the noun (and adjective) "1970's"; is it plural (the years between December, 1969 and January, 1980), or is it a genitive, symbolizing the seventh decade of the twentieth century, beginning with the (nominal) year, 1970? Think of the 1970's versus 1977's hit movie, Star Wars. If the former, then the apostrophe is incorrect; if the latter, the apostrophe is correct. In a synthetic language, these choices are overt and change the sentence; in English, which has stripped a lot of the nuance from such symbol-object representation, the argument is harder to follow (and needless in such a trivial example, but the reductionist approach does de-clutter the workspace).

Is a noun a possessive, a plural, or neither (or either and both)? [prolepsis]
Our verbal representations are difficult to prove against empirical reality, not because they are faulty but because of the nature of abstraction and the subjectivity of consubstantiation. Given this fact, how can writers square the circle of conveying a thought through an (unavoidably) incomplete system of linguistic semeiotics?[1] The answer is it cannot — not completely. So there will always be a compromise.

Epiphenomenal grammar
More interesting than the mechanics, methinks, is the process. Perhaps the Anglophone community — or some significant fraction, taken as an entity — re-presented[2] the linguistic token as a plural instead of a possessive? That is why I started writing numbers like 1970s, but perhaps I was misapprehending? (I did not consult an authority, I just changed my own writing style. I did complain to my peers, so there was some feedback.)

It would appear that a lot of us did (cue digression on the wisdom of crowds). But perhaps it was simply the urge to simplify. Why use an apostrophe if a reader will ken your meaning without it?

Whatever the cause, as Deacon surmized, grammatical adaption is emergent. (This has correlations with the wave-like (group) behaviour of the sum of particular human individuals; the phenomena of interference cannot adequately be explained using only the quanta of photons. Like the dichotomy of light — both particle and beam — humans are both individuals and their society.)

Grammar is important
I thought of a better example for possessive pronoun its and the contraction it's, being the interrogative pronoun whose (also used as a possessive: "the person whose book I have…") and who's, i.e., the contraction for both who is and who has. Whose brush is this? versus Who's (i.e., who has) a brush / (who is) brushing?

Note that in spoken English, owing to the symbolic power of the human mind and the trivial scope of this quodlibet, any association with "who" and "brush" will signify the person using the brush, hence this is a subtle (and generally unnecessary) distinction in semantics, but a necessary one for grammar to disentangle more complex thought, and understanding semeiotics.
Agreed; homophonic misidentification is the root of most grammatical mistakes; it is also the source of much comedy (including puns) and dissonant deep meaning (in oxymorons and paradoxes generally, e.g., the Liar’s Paradox[1]).

More importantly: how much does a struggling (English Literature) student cost to proof-read some text? You could probably get the proofing crowd-sourced for free. (Cue digression on the gig economy.)
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[1] Epimenides the Cretan said, “All Cretans are liars”; this is equivalent to “I am lying,”[3] or “This statement is false”; Douglas Hofstadter (1979), Gödel, Escher, Bach: An Eternal Golden Braid, identified this as a one-step strange loop.
[2] In rhetoric (the art of persuasion) this is the technique of paradiastole.
[3] This punctuation is now correct for both popular (US English) and proper (International).

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edit:
v1.02 footnote punctuation
v1.01 added missing footnotes.