Now I don't know a lot about Christian theology, but does it even matter that original sin didn't happen, so long as people sinned anyway?

Very much so, according to that theology. The point of original sin there is that it means nobody can be blameless, even in principle, i.e. all people are tainted by Adam and Eve's actions; opinions differ on whether this represents some sort of ancestral culpability or is supposed to work in a more metaphysical way. Either way, it leads directly to the idea that people need divine intervention to be absolved of that taint, which underlies the justification for the Gospels' sacrifice-for-redemption narrative and by extension pretty much all of Christianity.

(At least, that's the perspective I got from my Catholic relatives. Other Christian sects might have slightly different takes on it, but all the existing mainstream ones have similar doctrine.)