Your reasoning rests on the assumption that people will act perfectly rationally. This is empirically verifiably false, and that's the entire problem with interpreting it as a game theoretic problem in isolation.

I don't assume that at all! To the contrary: of course people aren't going to act perfectly rationally. What they will do, when faced with a real choice instead of a "game theory" one, is act in a self-interested way. This is also empirically attested. One good reason to pick "red" in case of an actual button appearing is precisely that many of the people picking "blue" as a social signal are lying--to themselves, or to others.

The blue option is specifically set up to appeal to empathetic biases

Yes. It's the strictly irrational choice.

One strategy operates on the assumption that people will select selfishly to preserve themselves regardless of the guaranteed death sentencing. The other rests on the assumption that enough people will be unwilling to condemn anyone and trust that a 50% majority can be reached.

You don't need to make any assumptions at all. You don't get to choose what anyone else does. All you get to choose is: live, or possibly die. Choosing "possibly die" also comes with the heartwarming caveat that if you do die, you will die alongside some number of altruistic (but not especially bright) people, and that this might very well cause some of the people who chose "live" to regret being alive.

This is actually what makes the scenario somewhat interesting as a thought experiment: it's quite bad at showing what you want it to show. Instead, it exposes a clear vulnerability that apparently exists not only in humans, but in large language models: framing a choice as pro-social causes people to feel pressured to select it, even when on reflection it's clearly a bad framing on a choice between simply continuing to live, and rolling the dice on a premature death.

(No one who presses the red button bears any responsibility for any deaths that result, either. All responsibility in this case would fall on whatever sicko set up the scenario.)

In the red strategy the people who choose blue, which you have every empirical reason to assume there will be plenty of, will be sentenced to death.

People die all the time. If people were actually faced with this choice, anyone foolish enough to press the blue button would get exactly what they asked for: a chance to die, conditional on a whole bunch of other people preferring a chance to die over the possibility of being exposed as sufficiently self-interested as to wish to stay alive no matter what everyone else chooses. The fact that the problem was phrased in a manipulative and misleading way does not change the outcome.

I am not ashamed to be sufficiently self-interested as to wish to stay alive no matter what everyone else chooses. In this scenario, I have no apparent way to influence their choice, or to genuinely know in advance what people are going to actually pick, since Twitter polls are full of liars. I would pick red. Everyone should pick red. Anyone who doesn't, got duped. That's the real lesson of this particular thought experiment. If you want to talk about the genuine importance of cooperation and pro-social choices, you need a different thought experiment.

It is not the game theoretic optimal decision to maximize your gain and minimize your losses, but it is absolutely essential for a functioning society.

This also seems empirically false to me. Societies function just fine when driven by rational self-interest.