Professor David Crystal explains it in his book The Fight for English: How language pundits ate, shot, and left (Crystal 2006), pp. 134-135:
Its is just as possessive as cat's, but it doesn't have an apostrophe. Why not? Because the printers and grammarians [of the nineteenth century - Alex B.] never thought the matter through [emphasis mine - Alex B.]. They applied their rule to nouns and forgot about pronouns, thus creating an exception (along with the food is hers, ours, yours, theirs) without realizing it. And even if they had noticed, they wouldn't have done anything about it, for it's was already taken, as it were, as the abbreviation of it is.
I highly recommend this book to anyone interested in languages.
Charles Fries (Fries 1927) speculates that it could also be so because "their plural forms do not end in s" (cf. one - one's - ones or other - other's - others, ft. 7)
For an excellent summary of how the rules concerning apostrophe use developed, see
Sklar, E. (1976). The Possessive Apostrophe: The Development and Decline of a Crooked Mark. College English, 38(2), 175-183. doi:10.2307/376342
If you want to learn more about how the grammarians of the past arbitrarily imposed their confusing rules - and didn't stick to them - see pages 197-198 in Doctrine of correctness in English usage, 1700-1800 by S.A. Leonard (Leonard 1962); for instance, Joseph Priestly argued in The rudiments of English grammar, which was published in 1772, the following (pp. 86-87):
On the other hand, in the same book, on page 11, he lists all the possessive pronouns without an apostrophe and he treats its separately, as the genitive form:
Thirteen years later, J. Mennye in An English grammar ; being a compilation from the works of such grammarians as have acquired the approbation of the public [...] argued diametrically the opposite of the convention earlier proposed by Joseph Priestly.
But in 1823, T.O. Churchill says the following in A New Grammar of the English Language:
An apostrophe generally indicates an omission, but in this case I would favour shan't over sha'n't for readability and consistency.
Sha'nt looks wrong, and I've never heard that their should only be a single apostrophe at the first omission.
Looking for a "general rule", an article called That Cute Li'l Ol' Apostrophe that claims:
We never use more than one apostrophe to a word.
While the general rule is to use the apostrophe in place of the last missing letter, such as in "shall not -> shan't", if we need to choose between missing letters that we'd normally pronounce and those that are silent, use the apostrophe to denote the missing sounds.
Another claims the following:
Well. at first glance, it appears to me that our way of ‘making’ contractions is to 1) lop off the last half of the first word and 2) smash it together with the “not”, contracting the “o” with an apostrophe. See for yourself…
shan’t= shall (minus the “-ll”) + not (minus the “o”) = sha n’t
Which is then moved together (sha->n’t) to spell: shan’t. Personally, I believe that this contraction (judging by the way we use the word, and say it) is an ‘evolved creature’ from the two contractions “shouldn’t” and “can’t”; as opposed to its parentage being shall and not. It just makes more sense. [Shouldn’t + can’t= shan’t]
The OED lists both shan't and sha'n't as colloquial contractions of shall not, but not sha'nt. Shan't appears in 139 quotations, sha'n't appears in 25, and sha'nt is in only five.
Project Gutenberg's out-of-copyright books are usually older and don't necessarily reflect contemporary use, but searching their August 2003 CD of 600 ebooks: there are 589 results in 103 books for for shan't, 122 results in 29 books for sha'n't, and only three results in two books for sha'nt.
Another common contraction, won't, comes from woll not (an archaic version of will not). It also has two chunks of letters omitted. Should this be wo'n't or wo'nt? Motivated Grammar writes:
Did the contractions won’t and shan’t spring into English fully formed, like Athena from Zeus’s noggin? No, interestingly. Samuel Johnson’s Dictionary (printed in 1855), has wo’n't, as do some (modern) editions of Lewis Carroll’s Alice in Wonderland and The Ohio Educational Monthly in an article from 1868. Likewise, sha’n't was commonplace in the old days, plastered across the pages of the dreadful Victorian novels that I had to read in AP English as a lesson as to what happens to those who show an interest in reading. Books like Evelina; or, The history of a young lady’s entrance into the world (why did every single book in those days have to have a subtitle?)
Now the interesting thing is that won’t and shan’t live side-by-side with wo’n't and sha’n't in these old books. Some quick results on Google Books between 1600 and 1800: 777 won’ts, 57 wo’n'ts; 216 shan’ts, 73 sha’n'ts. Between 1600 and 1700: 48 won’ts, no wo’n'ts; 1 each of shan’t and sha’n't. So it seems it was never the case that the multiple-apostrophe form was more common. For some reason or another, English writers have always preferred a single apostrophe over strict application of “put apostrophes wherever a letter’s missing”. (Michael Quinion guesses that the double-apostrophe form was a later edition, suggested by logic-minded grammarians, that died out because it was a pain to write and looked weird.)
Quinion pointed out shan't is actually older than sha'n't and summarised:
The abbreviation, as you say, strictly demands the extra apostrophe, and it was probably the influence of logically minded eighteenth-century grammarians who persuaded many people to put the extra one in to start with — but whenever did logic ultimately matter in language?
So: the rules aren't really clear, but shan't is the most common, sha'n't is somewhat old fashioned, and sha'nt is extremely rare.
Why isn't English consistent?
Because it's evolved from a big mish-mash of several other languages over some 1,500 years.
Why isn't Winnie the Pooh mandatory reading for all English speakers?
Because nothing is mandatory reading for all English speakers. If Winnie the Pooh is mandatory reading, what else should be mandatory? Where do you stop? The only mandatory reading for English speakers should be the English language.
Best Answer
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.