A friend and I
Languages evolve. If people say “me and X” over the formal “X and I,” then the more correct answer is “me and X.” The only people I’ve seen quibble over this matter is English language teachers in school.
Yes, language evolves, but this case is a bad example of that. To me, and other people I know, "X and me went to the movies" sounds exactly as wrong as "Me went to the movies." This is not an attack on people who are saying it "wrong", because I understand that it doesn't sound that wrong to everybody. But the fact that some people do hear it as I do is information that can serve a person's decisions. It's OK to not want to "sound dumb" in a specific context, even though it would be unreasonably mean to make that accusation. The opposite is also OK: To not bother adjusting for that audience - maybe you just don't care enough about the way they perceive language.
My point is, even though getting it "wrong" is very common, this grammatical rule is still far from dead in practice, and so simply offering the correction doesn't necessarily mean the person is being purely pedantic.
Of course this is all to do with "me" vs "I". I agree the strict order of the nouns is indeed almost dead in practice.