Legacy: "Play" for "Existing User" never works

Usually when the launcher comes up, it automatically logs me in just fine. Sometimes, however, it pops up this screen that tells me the following:

You've already logged in as theosib.
You may use this account to skip authentication.
Existing user:
[theosib] [Play] [Logout]
Alternatively, log in with a new account below:
.... etc.

If I click "Play," it never logs me in. It always asks me to type in my username and password. The next thing to pop up says:

We couldn't log you back in as theosib.
Please try to log in again.
(InvalidCredentialsException: Invalid token.)
....

When this happens, there's usually not been enough time for any time-limited authentication to expire. I can come back the next day, and it'll just log me in. I can come back in a few hours, and get the above to happen.

When I then enter my username and password, it logs me in just fine every time. This suggests that there is no connectivity issue with Mojang's account server.

One thing that has occurred to me is that there is this confusion over "email address" vs. "username." Apparently, it used to be that people would login as a username, and that's still valid for older accounts. My account is relatively new, so I'm supposed to login using my email address. But the "already logged in" screen shows my username instead of my email address. Is that part of the problem?