I've always said "accidentally", rather than "by accident". To my ear, "by accident" sounds almost as archaic as "by your leave". Maybe "on accident" comes from "on [making an] accident".
I almost never hear anyone say'on accident' but If I had to guess, it comes from taking sort of a parallel antonym to 'on purpose'. Did you do it 'on purpose'? No, I did it 'on accident'. It sounds weird to my ears but there is some kind of sense to it.
Maybe. But if I follow that path, I get "accidentally" vs "purposefully". The first is about how it went down, but the second is about how it was carried out. The opposite of "purposefully" is "carelessly" or "recklessly", no?