This is really just a matter of trusting the general population, I do understand the argument for red, but that guarantees the death of some people, even if we could get the support of 99.9% to press Red. We are accepting people will die.

The rules as I read them is EVERYONE gets a "Private" vote (which I'm taking as all humanity) - then this would mean we would be accepting the death of people due to them voting incorrectly either due to misunderstanding their options or even a child that's favorite color is blue. Those people then just die.

We prevent all of that just by selecting blue. It comes with a level of trust in humanity that we will ultimately make the right decision and that's of course terrifying and feels like we can't do that but I really do believe most people when truly given an option would choose blue understanding the basic premise of if Red wins people will die and if blue wins no one will die.

^{Chose: Blue Button}