The one that people get wrong all the time is "its" vs "it's" for exactly this reason. By the usual use of posessive apostrophe you would expect "it's" and yet that's only ever correct if you're eliding something (i.e. you mean "it is", "it has" etc.)

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 ;)