I rrplaced the whole drain assembly. Took about 40 mins...much easier than I expected. As I suspected, I was able to look inside the old drain once I removed it, and that bottom polished piece was threaded on the inside, but I can't get it to budge a single millimeter. I think the original installer used something (unnecessarily) that hardens on the pipe threads to seal it, and over the years it dried and racked enough to leak but not enough to let me unscrew it and see what was going on.
At any rate, the new piece is in and is leak-free.