Podcast Hosting

I'm afraid not. The copyright in the music in Apple Music lies with the record companies, not with Apple. However there is (independently of Apple) copyright free music you can use from various sources : see https://www.buzzsprout.com/blog/free-music-for-podcasts, and Google 'Podsafe'.

If you want to leave Anchor, check whether you have obtained a feed from them and submitted it to Podcsasts connect yourself, or whether you have the arrangement where Anchor submit it themselves - the latter is simpler for you but as far as Apple are concerned the podcast is theirs, not yours. You would have to check with Anchor whether you can wrest the podcast from their grasp.

This is a user to user forum: you are not talking to Apple here. And, as I say, Apple don't own the copyright of the music in their library. It's licenced to them for use in their library (some producers don't allow streaming, only purchase). You would have to negotiate individually with the producers and copyright owners of each individual track, and getting permission, even if agreed to, would likely be expensive.

Short Answer is Apple will not allow it.

TL/DR - With Anchor you can only distribute those "podcasts" with music in Spotify. You can't put your RSS feed out to the world or Apple Podcasts where the majority of listeners are. And Since Apple does not host podcasts - this option is not going to be something Apple ever allows. Again not an option that is ever going to happen. Best to start a new podcast that does not use RIAA music - maybe one that uses music from pixabay or magnatune or ccmixter which can be used on podcasts that are on Apple Podcasts and all the other services.

You have received replies from the two biggest contributors to the Apple Podcasts forums that have told you all you need to know - and that is Apple will not support this. You can either take the advice / information above and do something constructive with that advice or you can ignore that advice. But please note we both tried to help you.