True, because the pixels are larger and the standard deviation of the photon count is proportional to the square root of the count.
If you have a larger sensor with the same pixel count, the shot noise per pixel is lower.
You can use software (noise reduction and downsampling) to trade all that extra resolution for much lower noise in the overall image with the same sharpness (resolution), and in fact that's what you end up doing when you compare the two images at the same final size.That's something else again. This discussion was related to the discussion regarding the intensity and total quantity of light and whether how that was affected (if at all) by the size of the sensor [before post processing].
All those photons that are collected by all those extra pixels count in the total signal (sharpness) to noise (noise) of the final overall image, and that's the reason that a larger sensor out-performs a smaller sensor in low-light despite having the same sized pixels.No, those extra pixels don't count (again, before post processing). Consider a FF sensor with pixels that are the same size as an APS-C, taking a photo of a uniform 18% gray background and that APS-C taking the same photo under the same conditions. There will be no difference in the shot noise between those photos.