Absolutely true. Parent may just not understand all the details. One thing that can be slightly confusing w/r/t copyright and recorded music is that the there's a rule that automatically grants a license to produce a new recording of a piece, without the original author or the copyright holder even being consulted. This is called a "mechanical license"; you can perform your own version of any musical work as long as you pay the fee and royalties for it.

That said, I believe there are still provisions for something like "moral rights" where the original author could sue to block publication.

The US doesn't have a general concept of moral rights, though many countries do. Is there some moral right specifically applicable to those compulsory musical licenses?

You're right -- I don't remember the right term for this case, which is why I put it in quotes. I just know that you can get the license automatically, and do the recording, but that the original artist has some means to try to block you releasing the recording.

Yeah, it's pretty confusing. I'm pretty sure the rules for performing covers are different from the rules for releasing recordings of covers, too. I think performance rights are more permissive.