I personally found it confusing and un-Mac-like that quitting the configuration app also now stops the Tailscale service. It was unfortunate to discover this while I was AFK.

My recommendation is to rethink it to work like apps like 1Password, Default Folder, Keyboard Maestro, Ice, etc., where I can always easily open a configuration app, but the service must be intentionally/knowingly quit via either the configuration app or the menu bar utility.

TLDR: Please separate the service from the new configuration app.

This was driving me nuts but at some point very recently they added a "Hide Dock Icon" to settings.