The one-shot case requires Bayesian thinking, not frequentist.

I disagree, but I am inclined to disagree by default: one of the themes that motivates me to post here is the idea that frequentist calculations are typically able to give precisely the same answer as Bayesian calculations.

I also see no trouble with wearing my frequentist hat when thinking about single coin flips: I can still reason that if I flipped a fair coin arbitrarily many times, the relative frequency of a head converges almost surely to one half, and that relative frequency represents my chance of getting a head on a single flip.

The answer I gave is the correct one, because observers do not gain any information about whether the coin was heads or tails.

I believe that the observers who survive would. To clarify my thinking on this, I considered doing this experiment with a trillion doors, where one of the doors is again red, and all of the others blue. Let's say I survive this huge version of the experiment.

As a survivor, I know I was almost certainly behind a blue door to start with. Hence a tail would have implied my death with near certainty. Yet I'm not dead, so it is extremely unlikely that I got tails. That means I almost certainly got heads. I have gained information about the coin flip.

The number of observers that see each result is not the same, but the only observers that actually see any result afterwards are the ones in either heads-world or tails-world; you can't count them all as if they all exist.

I think talking about 'observers' might be muddling the issue here. We could talk instead about creatures that don't understand the experiment, and the result would be the same. Say we have two Petri dishes, one dish containing a single bacterium, and the other containing a trillion. We randomly select one of the bacteria (representing me in the original door experiment) to stain with a dye. We flip a coin: if it's heads, we kill the lone bacterium, otherwise we put the trillion-bacteria dish into an autoclave and kill all of those bacteria. Given that the stained bacterium survives the process, it is far more likely that it was in the trillion-bacteria dish, so it is far more likely that the coin came up heads.

It would probably be easier for you to understand an equivalent situation: instead of a coin flip, we will use the 1 millionth digit of pi in binary notation.

I don't think of the pi digit process as equivalent. Say I interpret 'pi's millionth bit is 0' as heads, and 'pi's millionth bit is 1' as tails. If I repeat the door experiment many times using pi's millionth bit, whoever is behind the red door must die, and whoever's behind the blue doors must survive. And that is going to be the case whether I 'have the math skills and resources to calculate' the bit or not. But it's not going to be the case if I flip fair coins, at least as flipping a fair coin is generally understood in this kind of context.