You lost me on the image level noise, Neuro. It seems that an APS-C sized crop of the FF image and the APS-C image in this case would be identical. The number of photons hitting each pixel is the same and assuming the downstream operations are identical, what's the difference?
I don't agree with this "amount of light" argument. Consider a full frame sensor and an APS-C size sensor with pixels the same size as a full frame taking photos with the same lens at the same f-stop and the same distance from the subject. The signal to noise ratio for each pixel in the APS-C sensor will be the same as the S:N ratio as the corresponding pixels in an APS-C sized area of the ff.True, but the 2.56x greater area of the FF sensor will gather more total light. Comparing noise at the pixel level isn't the same as comparing noise at the image level.
Yes, if a FF pixel and a crop pixel are the same physical size (and technology) the individual pixels will be identical in terms of signal and noise.
If both images were shot with the exact same lens, the same settings, and the same distance, the central 40 percent of the FF pixels would be exactly the same as the crop pixels.
In real life, with your equal size pixel scenario, we would try to frame the two pictures the same, so that means either a 1.6X longer lens on the FF camera, or walking closer until the image filled the screen the same. Either way you look at it, that gives you 2.56 times as many pixels of equal quality on the target, so when you "normalize" the FF picture for the same number of pixels as the crop image, you end up with better quality pixels on the FF image. You are choosing between more pixels of the same quality, or the same amount of pixels but of better quality. There is no way for crop to win in that scenario.
In the real world, with the cameras Canon makes now, FF wins the IQ contest in all but one scenario... and that scenario is when you are focal length limited, can't move any closer, have a GREAT lens, and good lighting. Under those conditions (happens a lot with small birds) the quality of your crop pixels is fairly close to your FF pixels, but you have more crop pixels on target so you end up with a better image from the crop camera. Everywhere else, FF wins.
NOTE that I have left cost out of the factors.... cost will change the point where you become focal length limited, affect lens quality, and may even eliminate FF altogether. If you have $3000 and want to take pictures of distant chickadees, a 7D2 and a Tamron 150-600 is your best bet. If you have $25,000, a 1DX, a 600F4, and a 1.4X or 2X teleconverter will be the best.