I looked through the log files and I found the culprit - I had "Disable visual themes" and "Disable desktop composition" (desktop composition had no effect on this when enabled again) in the compatibility tab in the file properties of arma3.exe and arma3launcher.exe files (for increased performance - no Win Aero), but it's still weird, because I'm pretty sure that it really shouldn't cause this. When I enabled it again, mod selection started to work. I'll update the steps to reproduce to match. Thanks a ton, you guys are the best!
I can still paste the log files in case you want to read them.