Would it be better if youtube removed the free, ad supported version entirely, and only allowed paid users?
Your data is worth far more to them than a $13/month subscription fee. In fact, if you do pay it, the data becomes even more valuable, because you're now guaranteed to always be logged in. You're also likely to use it more to get more "value" out of your purchase, generating even more value (for them). Finally, you've also identified yourself as the kind of person that pays for things that should be actually free.
Worse than all of this, when you use Google (or any of these malware/spyware companies), thanks to network effects, you don't just pay for it with your freedom, you pay for it with some of everyone else's too.
You can take great advantage of them if you can resist the "payment".
It is a bit of freeloading. The product exists because there are multitude of humans that cannot help but "pay".
It’s not as though free users listening with the app in the background is somehow an additional marginal expense as opposed to them listening with the app in the foreground.
For music videos there are different licensing terms for listening vs music videos. So if they don't appease the licenser than their contract will be less favourable.
And of course ads will pay less for people who aren't looking (although his is technically lost revenue, not an expense).