I'm thinking something is wrong, or perhaps I'm doing things wrong.
I've used Terminal to set my boot-arg to nv_disable=1 (overwriting nvda_drv=1).
I then reboot and never again reach the desktop (which makes screen sharing impossible).
There are some misconceptions going on here.
To actually see something on the screen or have a desktop, SOMETHING needs to create that screen. The 2 choices are EFI driver, or Nvidia driver. If you have a non-EFI card and you turn off the Nvidia driver, then obviously there is nothing to create an image. So nothing will ever show up on a display connected to an Nvidia card with no EFI.
You would be better off removing the card and trying screen sharing with no card, I seem to recall that it creates a small, default desktop, but am not sure.
The cards that require Web Driver are basically all Maxwell, and the last Kepler cards that Apple has decided to ignore. Titan Black and GTX780 Ti.
As to why a 650 needs Web Drivers I do not know. It shouldn't require them but I believe that the real issue likely lies with monitor definitions. You might be able to flash with another 650 rom and find card works but that would be a lot of trial and error. We used to do an EFI 650 but went with 640 instead as it doesn't need power cable and thus more desirable for many, we no longer play with 650s much. I won't go in to much detail, but the thing that crippled Nvidia flashing for so long was Monitor Definitions. Nvidia driver does some clever footwork to hide this fundamental difference in these. It works hard to keep you from knowing that it is working hard. Some lazy coder at Asus drops a byte in Monitor Defs that Windows can gloss over, it sometimes results in bad OS X behavior.
It is likely that adding EFI to that 650 would solve most problems, though I understand that the cost may exceed the benefit.
I have spent last several days trying to fix a 3yr old LG Plasma that crapped out. Testing voltages, reflowing inductor coils, replacing MOSFETS, etc. At this point I would have been MILES ahead if I had just called Earl's TV Repair to fix the thing. But I didn't know that when I pulled the back off, was hoping it was just a fuse. Hours I won't get back.
To the OP, I would suggest treating this thread as a practical thing instead of theoretical. Create a disc image of a basic 10.10.3 install. Before recording this disc image, download the 10.10.4 update and the needed Web Driver to the desktop. Use Disc Utility to make a restore image. Use a Non-EFI card from Maxwell. Document the exact steps and how they work to get install to working 10.10.4 with Web Driver. Get out a piece of paper and a pen and write things down as you do it, what you did and what the result it. When they don't work, restore the disc image and start over. Treating this as theoretical is causing a lot of guesswork to come in. Once you find a method that works, do it again.