I was under the impression that
извинить - из-вин-ить - to forgive, pardon
извиниться - из-вин-иться - to apologize (by extension of the idea "to excuse oneself")

As you say, that is in fact how they are used. But I didn't realize that the etymology of the -ся in reflexives, like this one, was so "close to the surface" in natives' minds.

"I'm sorry that it (homework) is late"
which is sort of different from:
"Sorry, it's late."
But now that I think about it.... I can't tell the difference. Also I don't know how to punctuate either of these sentences because (had to look this up because I couldn't tell) "sorry" is a interjection, which ruins everything.

Anyway, this is a good question because English relies a whole lot on gerunds, so it can be hard to form sentences without them.