Transfer non-purchased music between iPhones?

So I'm switching between iPhone 8 and 13, and I've run into a problem: I can't seem to transfer music not purchased from the iTunes store to my new phone. I also own a MacBook Air, but of course that means its hard drive space is limited and I can't keep my entire music collection on there. Half my problem is how big this music collection is.

I first set up my new phone by restoring from an iCloud backup from my old phone, but none of my non-purchased music made it. Then I tried backing up the old phone onto the Macbook Air and restoring from there, but it not only didn't transfer the music, I also didn't get my playlists this time.

Is there a way I can transfer non-purchased music easily? This sort of thing seems to have fallen to the wayside in the era of streaming music. Google search results are mostly giving me ancient answers, and even the few newer ones from a year or so ago are telling me to change settings that don't seem to exist anymore. There's also quite a few results that are thinly-disguised ads for pricey third-party programs.