Core forum form dropdown empty
Right, so I tried all sorts of illogical things, one of which was deactivating and uninstalling Forum Access module (http://drupal.org/project/forum_access). 

I did not have to - because the form was accessible - but I did anyway, out of desperation.

Then I enabled it again. Set all permissions again.

And things work!

BUT this is not a proper solution. Something in Forum Access was screwed up badly and ideally this issue should've been solved at its root, not by a workaround like mine.