Songs from iPhone won't sync to or show up in iTunes on Mac

The Mac is using High Sierra (highest it will go). iPhone 13, latest iOS. I've NEVER had an issue syncing my music back and forth between the computer and the phone. For example, in December I sync all my Christmas songs to my iPhone. But in January, I remove them all so they aren't taking up space or showing up when I shuffle. I do this every year, no issues. I recently gave that computer to my wife to use and backup/sync her iPhone. I've done the sync fine, I've backed the phone up to the Mac fine, but when I go into iTunes ALL of the songs only show as downloadable - not actually synced to the Mac. It synced the albums, playlists, and the songs - but only the fact that they exist on the phone...NOT the physical music files to the computer. I could go through and tell it to download all of those songs via the internet, but that's silly and I shouldn't have to. I've checked all settings on the phone and the Mac, and nothing indicates for it to not sync the actual files to the computer.

Note: MOST of this music was purchased through the iTunes Music Store, but NOT all of it. A lot of it was created from CDs we physically own, turned into MP3s or AACs using the SuperDrive in the computer, and added to iTunes.

I also discovered that when I have the iPhone connected, and click on it in the iTunes window, it says there is ZERO music on the phone (0 songs). But that isn't true as I can go on the phone and see in settings how much space is being taken up by Music on the Phone (several GBs of actual files).

So in summary: there are physical music files on the phone, it syncs the *library* to the Mac, it does NOT sync the actual files back to the Mac, and iTunes thinks there is no music on my phone (even though it synced the phone library structure to the Mac). I do NOT have Apple Music. I do NOT have iTunes music match. This has always been physical files either created from our own CDs or downloaded from the iTunes Music Store synced back and forth between the computer and the phone.

Why does this matter? Why do I want the files on my computer (especially those we created)? I've started doing offline backups to the cloud and the first main backup is going on a hard drive they sent me. I'm doing an initial master backup that will be mailed to them to upload to their servers, then the software will daily keep any new files or changes on our computers backed up into the cloud. I want our music, and many other things, in this initial backup.