I'm pretty sure VLC (from f-droid) does what you want. It might prefer to have full file access, but you can just deny that and just open files from a file browser or whatever.
Is it working for you? It is not working for me, it keeps demanding permissions. It might be something specific to my phone, but also unlikely, since there are video and audio players that can play videos and audio without any permission at all.
Just be aware, that music player without file access won't be able to traverse/list directories, so playlists won't work. This is a limitation of any file-picker based file access (e.g. flatpak applications on linux, accessing files via portals have exactly the same limitations). As the sibling comment says, it is possible to run vlc this way, but be aware of the limitations.
I don't have "2 buttons" on the screen titled "Permission not granted", and no file dialog. Also not working, when asking VLC to play the file from another program. Must be a bug or incompatibility somewhere. Also this must be the reason why I wasn't able to find a music player on F-Droid then :| Thank you for checking!
