Well, well. I decided to do an entire fresh manual installation while I was trying to figure everything else out. I did it on my laptop. Evidently, the newest download I'd done on that machine (since there's no indication which version on the downloaded folder) was the one just before 3.1 -
So I just simply went back in time, and now I've got my visual editor back, and i'm very happy. Never solved the problem with 3.1, but everything works now - and I'm not sure I'm really missing anything else that I had with 3.1.
Hope I'm not stuck here forever. But my functionality is back. And that's what I need right now. I hope they fix it, whatever it is.