I got it working (with substantial tinkering, but the game is worth it). I used the install script from lutris.net, tweaked it to use a hard-coded prefix, switched from fullscreen mode to windowed (to avoid having to hard-quit the X server in case the game crashes) and pressed ESC on the company splash screen. I guess the most important part is pressing ESC on the splash screen, but I'm not 100% sure. A ton of thanks to @GameOnLinux and @IronScyth for their support (on YouTube) in getting this working!

While tinkering with the installer, I switched to windowing mode to avoid having to reboot the computer in case the game doesn't start properly, and I never went back to fullscreen (too lazy).

When the computer enters sleep mode, the game seems to be crashing after a while (sound is still playing, but I can't get back into the game.