...has anyone managed to persuade GOG's Defcon to run on Windows 8.1? It either silently doesn't start, or brings up an annoying banner saying "this app can't run on your PC".

I just installed it with Windows 8.1 and it ran without any errors. I'm not using any compatibility settings to run it. It does work in 8.1. I'd contact support to help you sort this out.

Your post made me do the uninstall-install dance again (I'd already tried it). It worked. I have no idea why it didn't before. But it works now.
For anyone else with problems, try uninstalling and reinstalling. Just in case.
Now I can be rubbish at another game, too.