I'd like to understand why I can't use a song in one of my videos without permission/payment, but an AI company can train models using that song without having either.

You're right, it's an unjust situation. And you may note that no one else besides the AI companies has made any progress at all towards changing it.

Copyright will soon die, having outlived its usefulness to society. Whether the knife is held by someone named Stallman or someone named Altman is of little consequence.