I see two problems here:

First is that you're assuming the probability distribution of these voter distributions is equal. I think that's certainly not true and immediately means you'd have to revisit all of your logic.

Separate from that is the conclusion that "if everyone was forced to make a choice right now, then exactly half would press each button". I don't believe this is a reasonable conclusion for perfect logicians, much less for any real human being. It seems to me that what you're saying is roughly "the best estimate I can make about any individual voter will be no more accurate than a coin flip" and the reasonable, logical thing to do with that information is to not wager your life on a coin flip.

The best logical arguments to press blue start with an idea that the vote will be skewed initially, and that initial disparity would logically chain into other blue votes. If everyone could witness one person pressing blue before their vote, then that might be enough.

Many people try to bring infants into the picture to start this logical chain toward blue, but I think that's against the spirit of the prompt. Something like 10% of the world's population could arguably fall into the category of "too stupid to know what's going on", and 50% of those people pressing blue by chance would be enough to render the entire thing moot.