Huh. It works fine for me. However, sometime around then it stopped launching the bundled browser & started launching the system browser to talk to the webserv

Huh. It works fine for me. However, sometime around then it stopped launching the bundled browser & started launching the system browser to talk to the webserver that runs in the background & manages the local storage (SQLite IIRC) and communications with HMRC themselves. So my first guess, if it isn’t working for you, would be to test whether xdg-open does the right thing on your system. A year or two ago it stopped working entirely due to a libc mismatch, which I had to fix by patching in a more recent libc. That was less fun.