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Choosing n to be 100% of the population seems like it should be perfectly acceptable considering the limitations on natural experiments I don't see how it can be argued that it is wrong. Ultimately n is chosen so that the bell curve distribution makes 2 sigma around 95%. But if you sample the entire population the bell curve shrinks to a straight vertical line, its the actual solution after all.
It is generally easier with examples lets look at election polling, the first thing we pick is sigma 2 so we have a predefined look of the bell curve we want, out of a population of millions, tens of millions, billions, n = 1000 is still always valid for all because we are working from the bell curve backwards. So polls essentially say that the population is 95% likely that it is within +-3% of what they claim.
The numbers get trickier as we reduce n but as you said there are cases where you are stuck with it, there are other statistical methods I am not really familiar with, but my thought experiment is that if you do sample 100% of the population the bell curve looks like the population is exactly 100% within +-0% which is exactly a vertical straight line bell curve.