> 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.

Because training isn't redistribution.

You can also listen to the song and make a new one that sounds similar, just like the AI can.

To do that training, you must first obtain the item with the content you require. Did OpenAI purchase a copy of every book they trained their models on?

Answer: They did not. That is literally why there are dozens of ongoing lawsuits in progress.

For songs, it's not that hard to legally get access to it, I think. I'm not sure if Spotify can legally prevent you from using songs for AI training for example.