Well, I'm not sure what's going on with this problem, but I tried to log in again this morning, and was able to.
So I took the opportunity to try uninstalling again, and lo and behold, it worked.
I re-installed everything, including Playonline, as it seemed to be that causing the problem, and after the 2 hour version update finished installing, I can now log into the game once more. We'll see if it happens again or not. I hadn't done anything besides what I tried above, so I wish I knew what had caused it in the first place so I could advise others who might experience this problem.