Yeah it would but it would not route your traffic through internet if both devices can communicate with each other over LAN.
DNS have nothing to do with SSL. Tailscale provide routing. It does not change applications running on the server.
TS have some convenient features to set up SSL but you cannot choose the domain name freely. I bet you can purchase a domain which redirects to ts domain id you want.
It feels like those tiny bars should be stocked on top of each other with all the spof projects and services