I'm afraid that doesn't wash - if the short answer is correct then it 
should apply to other nouns, not just pronouns.

Compare "John's stealing Mike's umbrella", and "it's eating its lunch".

My question is not 'why does one apostrophe take precedence over the 
other', but 'why are nouns and pronouns treated differently'.

There is an assumption underlying your first answer that I'm not sure 
about; that aspects of the written language exist in order to distingiush 
similar forms with different meanings. The language is full of ambiguity 
which has to be decided by context (ever looked at the number of meanings 
of the word 'set'?), so why would "it's" and "its" have been singled out?