Sadly it seems there's a whole generation that's been raised without anyone correcting them on this specific point (that it's "Tim and I"). Not that this means I disagree with your greater point.
To be fair, a decent number of people also overcorrect and use "Tim and I" for the objective case as well (e.g. "The teacher told Tim and I that our grammar was incorrect"). I think simply giving the correct construction for the case that the speaker actually used isn't sufficient for using proper grammar; one needs to be able to identify which case is which to be able to use "correct" grammar.
The real fun is when someone corrects you for saying "The teacher told Tim and me" and then you give them a grammar lesson to the use of the objective case. I'll admit to a little bit of smugness when that happens.
Otherwise no, I don't correct people's grammar unless they've specifically asked me to review something.
Nicely put, I didn't realise it but I am, as you say, overcorrecting in this way myself. Interestingly I have a friend who has quietly tried to point this out to me (a counter example to the greater point) but you have explained it better.
Whether something is “correct” or not in this particular grammatical regard comes down to cultural factors, not objective linguistic ones. If we look at the languages of the world, we can see plenty of cases where the nominative form of the 1st person singular pronoun was gradually replaced over time by the accusative or oblique form. Therefore, people coming to say “Tim and me” instead of “Tim and I” is a completely natural linguistic change and complaining about it is simply being pedantic.
