Ok guys, here's the EXPLICIT solution to logging in:
Go to the page of a server that you want to join. Click on Show Players, click a player to go to their stats page. Click on the link to go to their profile. Click on Join game. Try to log into Origin when it pops up.
NOW THE IMPORTANT PART:
When you get an error, it will say that there was a connection problem. It should then have your password already entered for you. If your password is there, then DON'T TOUCH ANYTHING. It may work or may very likely not work at all, saying that your password is wrong. If this happens, then DO NOT CLICK ON LOGIN AGAIN. Close the Origin window, then try to join on a player again. Keep repeating this process. At one point it will just log in anyway.
Now, about your stability for staying logged in and your ability to join maps once logged in, that's a different story entirely. Have fun with that one.