A simple (low pass) anti-aliasing filter has a linear drop-off, meaning the greater the difference between the frequency of the signal and the frequency of the noise, the better the signal to noise ratio. On the other hand, sampling random gaussian distributed noise will be random gaussian distributed noise no matter what frequency you sample it at.
Remember that the lowpass will change the phase or time response of the original signal. If there is anything in the frequencies and phases affected by the lowpass it will get distorted.
The trick works only because the signal is spike like. I reckon it will miss some of the spikes in real life on real signals or produce spurious spikes. The statistics are not accurate at any given point in time just in the limit.
I was going to point this out (phase response distortion) but you beat me to it.
During my university period, I had to digitally sample an ECG (electrocardiogram) signal. Since the shape of the signal is important in diagnosis and must be preserved, I used a Bessel filter [1].
https://en.wikipedia.org/wiki/Low-pass_filter#/media/File:Bu...