I think the problem is because CRT displays didn't have pixels so the uniform noise which is static was not only uniformely spread in distribution and intensity (i.e. greyscale level) but also had "dots" of all sizes.

Also another possible thing that's off is the speed at which the noise changes: was it the 25fps refresh rate of a CRT monitor, related to that rate but not necessarily at that rate or did the noise itself had more persistent and less persistent parts?

The noise is basically the product of radio waves at all frequencies with various intensities (though all low) with only the ones that could pass the bandpass filter of the TV tuner coming through (and being boosted up in intensitity by automatic gain control) and being painted along a phosphorous screen (hence no pixels) as the beam draw line by line the screen 25 times per second so to get that effect right you probably have to simulate it mathematically from a starting point of random radio noise and it can't be going through things with pixels (such as 3D textures) to be shown and probably requires some kind of procedural shader.