I just tried downloading my family tree as a GEDCOM file, and it worked they way it has always worked. You tried most of the things I would try already (logging out and back in, then a different browser). I think the next two things I would try are:
Try from a different computer.
Create a new family tree (don't delete the old one) and try that tree.
It's also possible that it was a temporary problem, and will work now. Best of luck.