Alternatives to the Music App on my new Mac, to play my own music collection

A suitable solution here could be Elmedia Player. It has a folder-based structure, similar to Finder. Also, unlike Apple Music, Elmedia does not require you to upload or sync your files to a cloud service - if you've ever faced this issue, you realize how great it is.

You didn't mention formats or playback quality, but I'll note that it's great in that regard, too.

Next solution - VOX Player. It provides excellent playback choices and FLAC support, although its organization capabilities may not be as extensive as Elmedia player's. But it's worth a shot.