I use the same distro to test the game before uploading so it should be fine, that's with an AMD gpu though. My guess is that there is an issue between the intel gpu driver and the game engine.
According to wikipedia the gpu you listed supports Vulkan 1.3 under linux so that should also be fine. If the gpu didn't have vulkan support, Godot should display a popup message letting users know.
If you run the game from the terminal, it should contain some .net error messages if the game crashes. They might shed some light on the problem.
The game appearently requires a newer version of GLIBC than the one included in Linux Mint 21.3. You can probably update it or perform a system upgrade with apt.
Ran it through terminal. Any idea what this means?