What is Won’t a Contraction Of? – Etymology and History

This is covered in the Cambridge Grammar of the English Language (CGEL), as it turns out, in Chapter 18, “Inflection Morphology and Related Matters”, section 6, “Phonological reduction and liaison”.

The form ’s, representing either has or is, along with ’m (am), ’re (are), ’ve (have), ’ll (will), and ’d (had or would) are called clitics, and they are a variant of what are known as weak forms of words, which are pronunciations of words like a, have, from, you, etc. (about fifty in total) with a reduced vowel, such as schwa.

In the discussion of weak and strong forms, CGEL points out that there are certain grammatical contexts that require strong forms, and one of those contexts is something called stranding, where the object of a phrase is preposed (moved before the phrase). These are examples they give of stranding requiring strong forms:

a. Who did you give it [to __ ]?
b. We’ll help you if we [can __].
c. They want me to resign, but I don’t intend [to __].

In each of these cases, the word in the brackets has a weak form, but it cannot be used in this context because its object has been stranded. Of course, in written English, there is no difference between weak and strong forms—it’s only a spoken difference—but clitics are distinguished in written English, and the restriction on weak forms also extends to clitics. (There are additional restrictions on clitics, but they are not relevant to this discussion).

So, thus we can say that the second is in the sentence It is what it [is __] cannot be reduced to either a weak form or to a clitic because of the restriction to strong forms in cases of syntactic stranding.

It's not about a contraction "winning" over a possessive. "Its" is the possessive form of "it", like "his" is of "he", "her" is of "she" or "their" is of "they". There is no missing apostrophe; the forms go back to a time when English was a highly inflected language. It predates modern, or even Middle, English.

The possessive formed by the apostrophe+s construct is a more modern, uninflected, less-marked form. There are only a very few commonly used words — pronouns — that still use the older forms. Markedness tends to survive in words that are used very frequently, even when other aspects of the language are losing their markedness. It's the same reason why we still say "men, women and children" rather than "mans, womans and childs" when the plural ess marker is nearly universal in the rest of the language.