If you're comfy with stuff like that, you may have some luck forcing the resolution/ full-screen mode via the commandline:
-popupwindow Create the window as a a pop-up window, without a frame.
-screen-fullscreen Override the default full-screen state. This must be 0 or 1.
-screen-height Override the default screen height. This must be an integer from a supported resolution.
-screen-width Override the default screen width. This must be an integer from a supported resolution.
-screen-quality Override the default screen quality. Example usage would be: /path/to/myGame -screen-quality Beautiful https://docs.unity3d.com/Manual/CommandLineArguments.html
If true fullscreen is a problem, a popupwindow with full resolution is visually the same.
I'm comfy with command line, but there's a catch. If you use bundled "start.sh" from GOG, i.e. "./start.sh -popupwindow", the argument is not delegated to the game, and it starts again with black screen + mouse cursor.
I simply edited my own "my_start.sh" with trivial content "game/ChaosRebornLinux.x86_64 -screen-fullscreen 0 -popupwindow -screen-width 1920 -screen-height 1080" and this works for me, it starts the game visible.
So the conclusion is:
* the latest version on GOG as of today (Patch 1.13.3 16 August 2018) is still unable to correctly switch to full screen on my machine
* you can use command line options, when you avoid the GOG starting script (makes me wonder what good is it for then, and why they can't make one with argument support) ... (and why there is 32 bit version included in the... oh wait, this is not 64b deb package from repository... it's so weird+cumbersome to install/update/stay-aware anything without the package manager ... anyway I wonder who's using still 32b OS now after like 15 years of 64b availability)
This workaround should make the game run, if you have the same issue (not similar, like corrupted graphics, but same one = black screen with cursor mouse only), and I guess this is pretty much final state for the foreseeable future.
(as a SW (and ex-game-dev) programmer I'm sort of curious what's so wrong about Unity vs libc that they can't fix it on engine side, but whatever...)
Anyway, thank you for all the support and advice. :)