Thanks for the info and thanks for the offer! Actually I just now uploaded a Linux version of the demo, if you are interested to test it. Although I have only tested that the game works with wireless Xbox Series S/X controllers with the USB dongle (I'm using Xone drivers under PopOS). I still need to verify that other game controllers work also. Are you still using only keyboard? Or do you perhaps have some game controllers?
The AppImage could be actually a good way to package the game in the future. Currently I just compressed the game to a zip file (the executable file is named "T-Time.x86_64"). I will have to read more about the AppImage (how to make an AppImage).
The user auto updatable might be bit out of my current scope. And it seems also that in Flathub and Snap Store, you can't distribute any paid games currently. I'm not sure if they allow to distribute a demo of a game in those.
(There's a small bug currently in the Linux version with Xbox Series S/X controllers that the Menu button in the controller doesn't work as a start button. You have to use the View button in the game controller for start)