After the file integrity check the window goes away. I can see "setup_outcast.exe" and "setup_outcast.tmp" as processes on the task list. Nothing else seems to happen.

Seems that it needed admin consent to run, and the UAC consent program needed to scan a lot of files or something. Eventually, I got permission request, and then it did ANOTHER file integrity check, and then I got the actual installation panel.

The problem was that consent took so long I would try to run the installer again and it spawned even more consent processes which began fighting over disk access and slowing the process down even more.

Yes, the version 1.0 GOG installers do this. If you run them as a standard user the integrity check runs then it asks to run as admin via the UAC prompt which then causes it to re-do the integrity check again (though you can skip it). Best thing to do is to run the GOG installer as admin by right-clicking and choosing the 'Run as administrator' option.

The new version 2.0 GOG installers don't have the integrity check, they are instead digitally signed so you can check that the signature is valid from the file properties of the executable (however, multi-part installers still use the file integrity check as the signature can only work on single files). They also kick off the elevation to admin at the start (though I've not tested this on a version 2.0 multi-part installer). I'd still recommend running them as administrator though.