huntgmat
But public is not mentioned in the sentense .....
How can "it" refer to public ?
For me,
public's widespread belief Public's + widespread are adjective of "belief" , isn't it ?
For example,
Harry Potter's movie is good, but its for children.
its can never be Harry Potter ???

................. help .... SOS
Hope this helps...
It's is a contraction for it is or it has.
Your example of "its" in the Harry Potter sentence should say "it's". ...but it's for children is same as but it is for children. Its is a possessive pronoun meaning, more or less, of it or belonging to it.
So in the question above "its" refers to the public's belief. And there is absolutely, positively, no such word as its'.
A simple test
If you can replace it[']s in your sentence with it is or it has, then your word is it's; otherwise, your word is its.
Another test
Its is the neuter version of his and her. Try plugging her into your sentence where you think its belongs. If the sentence still works grammatically (if not logically) then your word is indeed its.
Examples
It's been good to know you. Contraction: it has
It's a bird! It's a plane! Contraction: it is
The dodo bird is known for its inability to fly. Possessive pronoun: its inability = the dodo bird's inability
Source:
https://www.stormloader.com/garyes/its/#top