In case anybody has the same problem I did: The problem turns out to have been much dumber, but also much easier, than I feared.
I finally pulled the speaker to replace it with a new one last night, and it turns out the contacts going out to the speaker were pried open on one side; when I would turn, the speaker plug would sometimes touch both sets of contacts and turn on. I pushed them back together and everything sounds—well, not good, because it's the stock speakers, but right.