I disagree in part.
To begin with, I presume that we really talk about prints. Whenever you print the image will be resampled to the native resolution of the printer driver. The resolution of the file may matter little, so far signal is dominated by shot noise only the total number of photons captured will matter and that number is not affected by binning.
Shadow noise can be dominated by readout noise. Lets do some very simple math:
Full well capacity is 65000
Nominal ISO is 25
So 1600 ISO = ln(1600 / 25) / ln 2 means six stops of underexposure. Let's also assume that we look at 12.5% grey tone, 12.5% is 1/8 or three stops.
So our grey tone will hold about 65000 * 25 / 1600 / 8 = 127 photons (well electrons to be correct). This will result in shot noise of 11 electron charges.
No readout noise on the DALSA chip used on the IQ180 is about 12 electrons (according to data sheet), noise is added in quadrature, so total noise would be:
sqrt(11*11 + 12 *12) = 16.3, leading to an Signal Noise Ratio of 127 / 16.3 = 7.8
Now, if we subsample four pixels the math will change a bit:
FWC = 4 * 65000
Electron count = 507
Giving shot noise of 22.5
Readout noise will become sqrt (4 * 12 *12) = 24
Total noise will be sqrt(22.5 * 22.5 + 24 *24) = 32.8 giving SNR of 15.4 (about twice the SNR)
If we use hardware binning everything will be the same, except that readout noise will not be added up but stay at 12.
So we get: total noise = sqr(22.5 * 22.5 + 12 * 12) = 25.5 leading to SNR of 507 / 25.5 = 19.8
So downsampling will improve SNR from 7.8 to 15.4. Binning in hardware will improve SNR to 19.8.
;-) This is the math ;-)
Keep in mind that downsampling will not improve image quality in print.
I'd also suggest that raw converters can take different parameters into account and do more ore less aggressive noise reduction. For instance, it would make a lot of sense to increase noise reduction in the darks, which often have impulse noise and probably little detail.
Down sampling in post in theory produces one stop gain in per pixel noise.
Pixel binning produces two stops.
In practice I find pixel binning to be a bit better than one stop difference; why? I could only speculate (maybe related to the modest ineffeciencies of demosaicing?)
But what do I know? I'm not looking at dXo charts, I'm only looking at raw files from said cameras in their native processor.
Downsampling has very little impact on noise. It might be worth pointing out, we are comparing two pixels with a difference in linear resolution of 2x (or area of 4x). The visual system will really not perceive any difference between the conditions is it naturally does its own downsampling cause by its own limit in resolution. Comparing images at 100% monitor view is not going to show the reality of the situation, unless you put instructions under your pictures for viewers to fix two different viewing distances depending if you downsampled or not.
Binning on the other hand is making a difference as it impacts the actual signal.