As weird as it may sounds, when your GalCiv3.exe gets terminated after splash screen, try running:

gc3_prefs_wipe.exe (in the game folder)

or when it won't help:

gc3_prefs_wipe.exe + gc3_font_setup.exe (Fonts folder)

It worked for me, and for my friend. We both had some GOG Galaxy leftovers from very, very early versions.
I believe new version of GOG Galaxy doesn't leave anything when uninstalled.

I had no previous install, and the game would fail either with or without Galaxy running. The fix was gc3_font_setup.exe - just that, and my problem went away.

I have 2 computers - a desktop, and a laptop that I am trying Galciv 3 on. The desktop installed and ran fine GOG version. The laptop has the very same issues as mentioned above. See the short video clip, and dies.

I've never had GOG Galaxy installed on either (unless Dell preinstalled on this new laptop). I also shut down my firewall temporarily, installed patches per Stardock website etc.

Prior to seeing this forum, I submitted a ticket to Stardock - waiting on a response.

Gonna try installing Galaxy, and see what happens.

An update. Firewall didn't fix it for me. Patches didn't fix it. Found an error as well regarding fonts, so I ran gc3_font_setup.exe (Fonts folder) as mentioned above, without installing anything else, and Galciv now works! The only issue I had is dealing with a 3K/4K screen and the fonts being so small - adjusted the resolution.

Thank you all for your posts! Very helpful to me!