Never mind...
I'm not very good at giving up. After finding QT DID work when I logged in as a different user, which I'd forgotten to try, I decided what the heck, and simply removed EVERYTHING Firefox or Mozilla related on my HD and then reinstalled Firefox. Since it's not my primary browser, I figured there wasn't much I could break (and I could also update the bookmarks to match Safari's into the bargain)
Bingo.
Quicktime now works.

I have no idea what was messed up, but something was, and it was something in my user account which has now been removed. I have no add-ons or extensions now either - not that I had any that I thought should interfere, but you never know.