Thanks so much for the info.
Unfortunately I can’t find anything to hint at why these particular animations won’t work. The ‘Select’ animations mode just shows an empty list, even if I press the Plus button.
I’ve tried making new animations but they don’t show up either.
As far as I can tell the only option I have is to trash this armature and start from scratch, remaking all of the animations. I don’t know how to copy animations from one armature to another, and I don’t know what will cause FBX Export to completely fail.