So the ASIC is essentially "screening" POSSIBLE candidates for a solution? It rapidly discards impossible solutions and only alerts the upper level processor when it "might be onto something"?
No - there's no 'screening', as that would presume that it would somehow be able to tell that some inputs are more likely to yield a valid hash than other inputs.
Keep in mind that the hash needs to be below a certain value. If you now look at the SHA2 function, you'll see there's several stages where you've already got a hash that exceeds that value, and don't need to perform any of the calculations after that point.