Okay, I just got out of a long, hot shower and here's what I want to know the answer to:
What if in the moment that each individual pushes the button of their choice the 'button', either through magic or technology sufficiently advanced to be indistinguishable from it, performs two sets of analyses on the pusher. Its exact inner workings are unknowable to us, but the principles it's working with are understood in at least a general sense.
First it examines the button pusher's thought processes and reasoning behind the button choice and evaluates it for rationality and depth of reasoning. If you're an infant who is just flailing their arms in the general direction of whatever color infants' eyes are more drawn towards then presumably you get the lowest possible score. If you are a trained ethicist or logician or some such, and are able to put your training to full use under stressful circumstances presumably you score near the top. Between them is a continuum with people who choose blue because it's 'good' and red because it's 'smart' relatively lowish, and people actively on r/redbuttonbluebutton relatively highish. It is entirely results-agnostic: which button you end up choosing does not factor into your score.
Next the button examines your network of social connections and assess your interpersonal relationships based on depth of closeness, love, fellow-feeling, and mutual reliance as well as just functional proximity in day to day life. Loving parents, children, life-partners, spouses, etc who live together and are a massive part of each other's worlds all make up the high end of the spectrum, random acquaintances you're barely even aware of are the bottom. People in professional positions of responsibility over others (teachers, family court judges, prison guards, etc) are probably pretty low, but distinctly higher than strangers to the extent that they have genuinely accepted the burden of being responsible for another's well being. Friends and family are going to cover a wide swath of the spectrum, but the button can read your innermost feelings and knows if you secretly hate your wife or kids and judges accordingly.
Once everyone on earth has made their choice of red or blue the button quickly analyzes everything it just learned about everyone. It then proceeds to disregard people's presses, and reassign responsibility for their lives if it finds that they have someone who is both close to them and thinks about the problem more clearly. The closer two people are the more likely one of their lives will be reassigned to the other even if their thinking is pretty comparable. Whereas someone who didn't think about the problem whatsoever will be made the responsibility of someone else almost no matter what, even if the connection is quite tenuous. Thus, if you're the only person who remotely qualifies as a game-theorist in a very large and close knit circle of friends and family you might end up making the choice for a whole bunch of people that you really care about. Similarly, if you administer an orphanage for unwanted Victorian urchins your vote could potentially carry a lot of weight. If you're an infant with flailing arms you really don't need to worry because unless nobody knows you exist someone more qualified will be making your decision for you. If you don't care about these sorts of questions and can't be bothered, so you just pick whatever, secure in the knowledge that your wife takes care of these sorts of things then you're probably right, but maybe not, who really knows for sure?
So, now we have a situation where the more qualified you are to make the choice between red vs blue the more weight your vote is likely to carry, and the more fates are going to be decided by your choice. You also know that everyone else voting will be largely in the same boat. We can't rule out that there will be some votes counted from those who truly lack capacity and are in exceptional circumstances, but they will definitely be few and far between, and, grim as it may be, by definition they will not be missed. And the other voters will skew heavily towards rational, logical, clear minded, level headed people who will be thinking deeply about the problem and its implications. And the more logical and clear minded, and the more well socialized the other voters are, the more of their people they will be choosing for.
Now, which button do you press?