You're missing the point, which is that the group with a lot more money to start have their voices heard more.
It matters some that their multiplier is different , but in absolute numbers its still more to the program that benefits fewer people. The "utility function" is not accurate because the wealthy's utility starts out with a massive advantage.
So yes, I think it would still be unfair if you switched it given the poor majority genuinely would rather have art than lead free pipes.
The problem is that their voices are counted less due to not starting with money.
But regardless, that would be a silly thing to switch because that's not a situation that ever comes up, while the original framing is a genuine problem in our society right now.