Doesn't that just mean that Github should be mad at them? Doesn't seem obvious that it actually gives me a license to do anything.

Pretty much all open source projects on GitHub rely on section D.6 of GitHub TOS (when you submit PR, it's licensed under project license and you have a right to do so).

TOS are not just fluff and paper, but legally binding. Granted, when there is a conflict between two conflicting requirements, it's never clear cut, but it's not just "GitHub will be mad."

Any project I've been involved in has its own explicit license, that allows such things (really allows much more, I don't touch anything closed source).

Is the ToS a contract? Even if it is would the remedy to Winamp breaking it be modifying the license Winamp provides?

Yes, every ToS is a contract. As for the most appropriate remedy, that's for a court to decide.