You're not alone in this. I too block port 53 on my network and block any DoH/DoT servers I can to force devices to use DNS through my pihole servers and those make calls out to predefined DNS servers.
I also agree that having a LOCAL_NAME_SERVERS would be helpful to do the initial resolution of whatever you're wanting and then use NAME_SERVERS after the tunnel is created.
For now I added my local name servers to the NAME_SERVERS list and while it takes longer to start the container it seems to work.