To me this is a weird use of "SNR" because you are viewing the sky background as a signal and read noise as a noise term. But the sky background signal behaves the same as dark current - and the signal itself is subtracted off leaving only the Poisson noise term as relevant. I guess you are ignoring dark current.
So - you are really just saying the exposure should be long enough that all noise terms dominate read noise - and I wouldn't use the term SNR here.
SNR has primary meaning in terms of the signal being nebulosity and the noise being all the noise terms that obfuscate it.
I will try to make a tool that shows how I view this stuff - which is different from how it is normally presented.
I am not really ignoring dark current, but with most of the cameras I use, dark current is so trivial by the time I've reached 10xRN^2 that I don't really care about it (i.e. it might be an electron or two).
To be more clear. The total shot noise in the LOWEST signal area of the image (counting ALL signal sources) should be 10xRN^2 for optimal results. It doesn't matter if that shot noise comes from dark current, background sky, or object signal, or any combination thereof. The key is that the total shot noise "swamps" the read noise. This isn't arbitrary, it is just mathematical.
If we assume we have 2e- read noise, then we would need 40e- signal to reach 10xRN^2 criteria. Now, we are measuring the WEAKEST signal in the image, regardless of where it may be. So your object signal, the signal you desire, should be even higher than this in other areas of the frame. But by measuring the weakest total signal (object+background sky+dark current), and basing your exposure criteria off of this, then you should have at least this good of performance throughout the entire frame, or better.
So, if we have 40e- background sky signal and 2e- read noise:
SNR = 40/SQRT(40 + 2^2) = 6.03:1
If we for the moment assume there is no read noise, then our background sky SNR would be:
SNR = 40/SQRT(40) = 6.33:1
If we calculate how good our SNR with read noise is, vs. how good a signal of pure shot noise would be, by following the 10xRN^2 criteria you will always achieve over 95% the SNR that you would if the camera had no read noise at all. Doesn't matter how many subs you stack, your "effective efficiency" or "stacking efficiency" would be 95% for the worst performing signal in the frame. Your total shot noise has a much greater effect on SNR than read noise.
Any area of the frame that does have object signal in it would perform BETTER than this, since you would have not only background sky and dark current, but also the added object signal. Exactly how much better depends, object signal could be zero or it could be many times that of the background sky. Regardless, by using 10xRN^2 you know that you have rendered the impact of read noise almost moot, so you don't really need to worry about it. Your performance is almost "ideal" (pure shot noise), and generally "optimal" from an SNR standpoint. (As I said before, I agree, there are other factors besides SNR to consider.)
Now, if you have 0 object signal in a given area of the frame, then your object SNR would be:
SNRobj = 0/SQRT(40 + 2^2) = 0:1
Now, lets say we have an area of the field with 80e- total signal. The object SNR for the pixels in that area would be:
SNRobj = 40/SQRT(80 + 2^2) = 4.36:1
If you had an area with 60e- total signal:
SNRobj = 20/SQRT(60 + 2^2) = 2.5:1
Yes, object SNR is ONLY the object signal divided by the square root of all the other signals and read noise combined. However, because we have swamped read noise, the object signal will hardly be affected by read noise:
SNRobjideal = 40/SQRT(80) = 4.47:1
SNRobjideal = 20/SQRT(60) = 2.58:1
This is why we talk about the 10xRN^2 criteria. The 40e- object signal is 98% as efficient as ideal results. The 20e- object signal is 97% as efficient as ideal results. No point in wasting time trying to do better than that, honestly. There may be a point to doing worse, but at lest you'll know where the upper limit is. (And, personally, I would consider 3xRN^2 to be the absolute lower limit, the bar-none, get this at the very least.)
If you can, or want, to achieve 10xRN^2, then it renders read noise effectively moot. Read noise has a minor impact to your SNR overall. No matter how many subs you stack. That is a useful thing to know, at the very least. Whether you always follow it, for every object, or not depends. Swamping read noise by less than 10xRN^2 will impact your SNR for a given integration. Again, SNR is not always the most important factor. There are other things to consider. And to normalize your SNR, you may only need to expose for slightly longer. But, at least you will know, IF you expose for Y with system X and achieve 10xRN^2, you'll achieve pretty darn optimal results.