So the game launches normally, gamepad buttons will cancel the logo screens, but once it gets to the title screen that says "press any button", nothing seems to count as pressing a button, so the game won't actually start. None of the buttons on my gamepad, mouse or keyboard have any effect, the game simply will not start and I have to alt+F4 to get out of it.

I have zero idea how to even begin to troubleshoot this.

This seems to be a GOG Galaxy-related bug. Starting the game with Galaxy active will run normally, but if you try to play offline with Galaxy off, then all button inputs are ignored.

I tried "Speed Braw" to play it with Wine 10 (using Xubuntu 25.10 STS; it is much
better than Wine 9 being part of Ubuntu 24.04 LTS after my current testings) and
got the same impression when I started the game as described above.

I contacted GOG support but got the info that there is no support for the game
for other OSs than the OS stated - but got an E-Mail to contact the developers.

Is the problem present when using Windows and not using GOG Galaxy
for a local multiplayer game? As that should work and then GOG support
may look to the problem.

I am now mailing the problem to the developers and will give further information
in case I get some from the developers.

If people experienced general problems with GOG Galaxy client and offline gaming,
I would be happy to read it here - so the problem could be important enough so that
it get fixed soon - especially as local multiplayer should work without DRM.