I've been searching all over the web (including the Adobe support FAQ and forum) for months now... and I just found the solution this morning. The trick is to disable Adobe's "GPU Sniffer", a program that attempts to identify the GPU and amount of vRAM present. For some reason it fails (now) if Photoshop isn't installed on the system drive. Renaming the file to sniffer_gpu.exe.old prevents it from running (obviously) and Photoshop then assumes everything is OK and uses the GPU.
Since this problem just appeared recently I think it most likely that there's a Windows update that gave it problems, but I'd definitely say it's Adobe's fault at root because they apparently wrote some code that assumes a hard coded path when it shouldn't.
In any case, I have it working again now. I hope this post proves to be of help to someone else with the same problem some day.