There certainly are songs about cock-fighting, and at least one about the 'running of the bull' at Stamford. I've never come across a song about bear-baiting. Perhaps that wasn't the same sort of communal activity that would inspire people to celebrate it in song?

The tradition is complex and frequently ugly and self-contradictory, but it is what it is. Why try to cover up its unattractive aspects, or
misuse it to support political ideas that were not even current when those songs were written and collected?

Personally I have no interest in hunting but no problem with singing hunting songs. It's the humorous ditties about wife-beating that I struggle with.