I understand that the video card has to go through the process to find the work itself. But couldn't that process happen with a little nudge in the right direction by the ASIC -- in theory?
No. There is no "nudge". You either have a solution or you don't. When you know you have a solution, you are done. If you don't know you have a solution, your work is of no use because it's millions of times more likely not to be a solution than to be a solution. The "work" is in determining if you have a solution or not.
The ASIC is 'dumb' in the sense that it's getting the answer through hardware bit flips at extremely fast speeds. It gets the answer, but has no work to show for how it got it.
The answer is the work to show for how you got it. The whole point of the design is that work is needed to produce the answer, thus the answer provides proof that work was done.
I'm merely suggesting that it might be able to pass that result to the video card, and the video card could make some use of it.
Until you have a useful result, you don't have a useful result. Once you have a useful result, you are done.