You wouldn't use its' at all-- the apostrophe goes after the S only if the word is a plural possessive, and it is singular. So basically all you need to remember is to use the apostrophe for the contraction, and not for the possessive.
I thought I had seen its' somewhere before.
Well, that greatly simplifies the problem!
letting things like this get to me seems useless. you can't correct/change the entire world, and trying to do so, even thinking too much about wanting to do so over something that trivial and global (compared to the truly disturbing things in this world), seems like a guarantee that you'll have a life-time of needless stress.
instead, i just make a quick objective mental note about individuals who make mistakes like these. much like i do about those who let it get to them to a high degree.