Agreed. But I'll note the shot noise is _approximately_ the square root of the gradient. Good bar graph in The Astrophotography Manual that illustrates the relative amounts. And, as you say, noise reduction in processing also can reduce the effects of the light pollution shot noise.
What infuriates me about Jerry is that he often pretends the small scale shot noise is the only noise worthy of consideration. Rhetoric that masks the reality.
Jerry is right, as small scale shot noise is the only true noise that actually affects the image. The gradient is technically (from a data/processing standpoint) just an offset, not a true noise (random time- and/or spatially-varying unknown deviations in the signal). The gradient is also pretty easy to correct as well unless you have a very complex field (i.e. almost anywhere in Cygnus.)
Shot noise is not easy to correct. Shot noise reduction also only affects the noise, and will not increase signal, and there is not necessarily a guaranteed improvement in SNR with NR...NR may simply improve the appearance of the image, but otherwise have no impact on whether faint details are visible or not.
Further, shot noise from LP is often by far the most significant source of noise in the image overall (depending on the sensor temp, dark current may be the only potential contender to LP for the dominant noise term.) LP alone can be dozens of electrons per second or more, can be tens to even hundreds of times stronger than object and background sky signals, so it has a huge impact on SNR. The gradient hardly matters in comparison. For the most part, the gradient can be dealt with via a change in offset, and there are many ways to model and remove gradients with various programs for more optimal correction. The loss of SNR due to LP is much more difficult to deal with. Noise reduction can smooth the noise profile out, however it will not actually increase the object signal, so there are specific limits as to what NR can do to help the SNR of a heavily polluted image.
What Frank is sharing here is that an LP filter may not actually improve SNR, and in fact may hurt SNR. For broadband objects, most LP filters (even the multi-pass type Frank is talking about) will cut out ~50% of the visible spectrum, if not more. That loss of light across the spectrum can result in a 50% (or greater) reduction in object signal, but only a ~30% reduction in noise. That hurts SNR instead of helping it. As much as signal grows faster than noise, if you cut out spectrum signal can also drop faster than noise. Depending on the kind of object, imaging without an LP filter may be better for SNR, because you keep more object photons, thus increasing SNR.
All of this really doesn't have much of anything to do with gradients these days, though. An LP filter does not eliminate gradients. Gradient reduction is usually necessary whether you are using an LP filter or not, so I would not call it an alternative to using an LP filter. And I definitely wouldn't trivialize the impact of LP shot noise, which if LP is present is usually the single most significant source of shot noise in the image overall, and thus has the greatest impact on SNR overall.
The only difference between these two images is the amount of LP. Same exposure, different locations. One is a red zone, ~18.5mag/sq" or around there, the other a green zone, ~21.3mag/sq" or around there. The SOLE difference between the two is how much LP each image has. Once the additional offset is subtracted away from the red zone image, the difference in noise is readily apparent:
The only difference here is LP. The gradient was easy to deal with. The remaining noise, however...was very difficult to deal with. Even after nearly 10 hours of exposure, the image remained quite dirty compared to dark site data: