No one says “me are going”, but lots of people say “me and X are going”, indicating that for native speakers, coordination of noun phrases functions differently than a single noun phrase.
In a contrasting case, people might think the phrase “I and John are here” sounds wrong, though they’d accept “John and I are here.” If you’re going by the “pure grammar” rule of “remove one subject from the phrase and check the grammar”, you’d have to accept either of those as equally viable, but many people don’t.