"It's" is one I've struggled with a lot. I understand "It's" -> "It is" but my brain wants to add an "'s" for possessive-ness. It just feels more right. I'm been able to mostly break that bad habit but I still don't like it.
You guys really need an Academy Anglaise indeed! (I wasn't aware that does not exist before your comment)
Honestly if we (in England) really had an institute of that kind we'd probably just end up formalising the weird spellings and grammar as being "right" instead of what we have now where our grammar rules are descriptive instead of prescriptive. Who knows how the differences with American, Indian, Australian, etc. dialects of English would be handled, but I'm sure we'd make a big mess of it somehow.
Edit: incidentally I now live in Sweden where there is such an institute, and they do seem (to my ignorant understanding) to make sensible updates to the dictionaries etc. to reflect actual modern pronounciation - but I'm still sure my homeland would figure out a way to mess it all up ;)
It was a major milestone for me when I made my first its/it's mistake in writing :)