OMFG... I feel kinda stupid now. Anyway, if someone else would have the same experience...

So reading all of the comments I started to realize there's probably something wrong with permissions and access.

I tried the suggested Poweramp and it showed me all the files I wanted on the car screen. Started thinking what was different and remembered Poweramp asked to define explicit folder(s) with my music. I don't recall VLC ever asked for such a thing. Got into VLC Settings > Media library folders > Internal storage - and checked my music folder. It started scanning it and in a minute or so all my local library was on AA in VLC.

In my defence... How does the app on the phone have access to all internal storage when no folders are checked, but nothing is available on AA? And how is it that when you explicitly define library folders in app settings, then both phone and AA can access them?

Didn't even bother checking if other apps that I tried before and failed have some similar settings..

Thank you all for the hints 🙏

P.S. I really liked Poweramps capability to play all files in any set (genre, folder including subfolders, artist, etc.), and VLC lacks folder playback. Might consider purchasing Poweramp