Another Update on this!
I found the root cause - bone orientations get messed up on armatures when they get flipped in 2.81 and newer, which in turn breaks the animations (since they are all releative to the orientations).
I've reported it as a Blender bug and it seems to be confirmed! With any luck, this will be fixed and the script will work with newer versions again.
If this turns out to be a bug, instead of a permanent change, back-porting the code to other 2.8X versions will be fine again. There will still be some trickiness for 2.79, but it is still probably doable too.