For me (running the game through Xbox Cloud gaming in a Chromium browser on Arch Linux; playing with an Xbox Series controller) when I get stuck in the title screen or notice in the mission that my right-stick axis are messed up, I just hit F5 (reload page → could also exit fullscreen and use browser controls to reload the page).
This re-connects me to the running session and the controls will usually start working as expected.
Obviously it would be much nicer if this would simply work out-of-the-box.