A cover is a transformative work covered under fair use. A similar argument to what you're saying could be made for any trend to ever exist on YouTube or the internet. No one owns the copyright for all the memes they make and share, yet I don't see takedown notices everywhere
The reason so many cover artists like Weird Al get explicit written permission every time is simply to keep them from getting sued (or to get any cases thrown out before having to hire a lawyer), but it's not legally required as long as you can prove your work is transformative.