Not installed via Gog, but via the backup installers. Quite a few months ago, I tried installing via Gog galaxy, and it just wouldn't download. I did get it running via the backup installers, but now nothing.
On Windows 11.
Can get to the screen that shows "continue .... exit" but none of the buttons do anything, have to exit via task manager.
Seems to be no entries in event viewer, no feedback of any kind as to where the error lies.

Have you tried to connect a gamepad? I just posted about an issue where my mouse doesn't work, but a gamepad does. Maybe it's a similar issue? I haven't found a fix yet though.
https://www.gog.com/forum/elder_scrolls_series/skyrim_ae_doesnt_recognize_mouse

i ran into this issue today (and then ran into the memory leak issue)

i needed to do an uninstall and reinstall and then the options menu came back and the play button worked

i am not sure what killed it - but it is back now

if it happens again i will see if i can fix it by replacing the launcher from a backup

i should note that i am now using the previous SkyrimSE.exe file (the one from the only earlier version available in GOG Galaxy) - since the exe in the offline installer is the latest and it has a memory leak (try going in and out of Whiterun a few times and watch your VRAM fill up and then have it start to fill your system RAM - on the previous exe that will not happen - and it does not happen on Steam)

anyway - hopefully you found solutions to this in the last 6 months - but if not then maybe some of this info may have helped :-)