As the title says. Starts up. I see it on my task bar, for a few seconds. I can't see the actual game, and then it shuts down.

Look through the threads on the game forum. https://www.gog.com/forum/disciples_series/the_game_wont_startwin_10/page1 is even stickied
Then also
https://www.gog.com/forum/disciples_series/disciples_i_ii_gl_wrapper_patch
https://www.gog.com/forum/disciples_series/disciples_gold_wont_launch_on_windows_11
https://www.gog.com/forum/disciples_series/does_disciples_1_and_2_both_work_right_out_of_the_gate_on_windows_10/page1

Running it as admin (and directly, not through Galaxy), would be the first step. If that's not enough, see if there's a config tool that you can tinker with. Try running windowed if you can. And also check compatibility settings, needs window manager disabled afaik.

Yeah, a wrapper is basically necessary at this point. You can use a generic one, like dgVoodoo2 for example, or any of the specific ones linked by Cavalary.