That condition imposes post-selection.

But not post-selection of the kind that influences the probability (at least, according to my own calculations).

Wrong - it matters a lot because you are using the wrong probabilities for the survivor (in practice this affects things like belief in the Doomsday argument).

Which of my estimates is incorrect - the 50% estimate for what I call 'pre-selecting someone who happens to survive,' the 99% estimate for what I call 'post-selecting someone from the pool of survivors,' or both?

You can't go the other way; we don't have any infinite sequences to examine, so we can't get p from them, we have to start out with it.

Correct. p, strictly, isn't defined by the relative frequency - the strong law of large numbers simply justifies interpreting it as a relative frequency. That's a philosophical solution, though. It doesn't help for practical cases like the one you mention next...

It's true that if we have a large but finite sequence, we can guess that p is "probably" close to our ratio of finite outcomes, but that's just Bayesian updating given our prior distribution on likely values of p.

...for practical scenarios like this we can instead use the central limit theorem to say that p's likely to be close to the relative frequency. I'd expect it to give the same results as Bayesian updating - it's just that the rationale differs.

Also, in the 1-shot case at hand, it is crucial that there is only the 1 shot.

It certainly is in the sense that if 'you' die after 1 shot, 'you' might not live to take another!