I'm gonna share something that I have kept to myself for a long time... here you go. Love you guys.
Yes, it works over the included USB-C to C without drivers, but i have a USB C dongle hub that i plug into a standard USB port on it and it works amazingly fine after driver install (even with VMware player). Oh, and no extra power cord.
It should work according to the monitor's info page. Your NUC supports DP-Alt through the rear Type-C port. Using a monitor like this, for example, would work as you've described. Did your monitor include a type C cable? If not, I'd return it as it's not working as described