This is newly installed on my laptop (Win10 Pro 64bit, i7 6700HQ, Intel Graphics 530, GeForce GTX960M 4GB, 16GB RAM, 250GB SSD, Intel 530). I had it running a while ago on a previous laptop that I think was running Win7 at the time, but I'm not sure about that - been a while.

I have this for Steam also and the same thing happens: when I start the game from either Steam or GOG (two different installs), the 3D video card selector opens with only the Intel Graphics 530 showing - although I'm not suprised it doesn't recognize the GeForce card. I click OK, the screen flickers a bit and then dfbhd.exe crashes.

I've tried the Win troubleshooter. I've set the properties to start as admin. On the Steam install I changed the compatibility mode to earlier Win versions, but for some reason I'm not able to change that for the GOG install - it's greyed out.

Has anyone gotten this to run on a current-gen machine? And if so, how?

The game should be detecting your Geforce card (detects my Titan X), and I'm guessing the game trying to use the Intel graphics is causing a conflict of some kind. Maybe try and find some way to force usage of the Nvidia card.

Edit: Running on latest Windows 10, Titan X, i74790k, etc.

That reminded me to check the game-specific settings in the Nvidia settings, but it was already set to for the executable to use the Nvidia card, so no luck there.

Could it have something to do with the fact that I'm on a laptop - I know some games to like to play nice with laptop video cards, but that is rare.

Laptops shouldn't be an issue for such old games, since they are way overpowered nowadays in relation with the games. This is an on-and-off issue, I guess some updates to OS or graphic drivers brakes compatibility. I had this issue multiple times, yet it got fixed eventually. I don't think it is easy to point out what is the exact issue with such old games.
And btw, I wanted to ask if these issues with Delta Force are fixed on the GOG version, looks like they are not, so there's no point in buying them again for the 3rd time. Too bad, money is money, maybe they just don't want more.