Tailscale routes my mobile device dns through my pile back at the home. I have nginx setup with easy to remember domains (photos.my domain.com) that work when i’m away as well without exposing anything to the open internet.
Why not call it VPN if that's what it is? In your case, it sounds like configuring your "pile" (is that a DNS server, short for pihole maybe?) on your phone would do the same thing, but if the goal is to not expose anything to the open internet, a VPN would be the thing that does that
