I was recently reading Outlawing Anthropics and I thought of a very similar technique (that one random person would be given a button that would change what everyone did). I think that it is a shame that this post didn't receive much attention given that it seems to resolve these problems rather effectively.

There probably could have been a bit more that justifies this argument, apart from the fact that it works. I think a reasonable argument to note that we can either hold the group's choices as fixed and ask about whether an individual would want to change their choice given these fixed choices, or give an individual the ability to change everyone's choice/be a dictator and ask whether they'd want to change their choices then. The problem in outlawing anthropics is that it mixes and matches - it gives the decision to multiple individuals, but tries to calculate the individual benefit from a decision as though they were solely responsible for the choice and so it double-counts.