I think you are right in theory, but it looks very different in practice. These automatic takedowns are done by the company hosting the files without any official legal regulator getting involved.

The rightsholder of such music licenses usually have vastly more resources and there are no negative consequences for false flags. This means it is better to overregulate which leads to a form of censorship.

I'm not a lawyer, but they need to make it absolutely clear that recordings that include copyright music shouldn't be copyright infringement unless the recording is solely containing the music and nothing else. Like if someone is talking over it, it should be automatically exempt. (Most pirates wouldn't talk over music just to avoid copyright laws. Most pirates are gonna want the original, high quality version, not one where there are people talking over it.)

I think such a change would be acceptable to most people, even those who are against piracy.