This is a popular fantasy, but I don't think it is true. People today are already voting for parties that increase tax on them and decrease it for others. Many people vote for things that are not directly benefiting them or are even bad for them, both consciously and out of ignorance.
I vote green/left, although they tax me higher, because I think climate change and social justice is more important than whether or not I have 1k per year more or less.
Given the amount of people arguing that the rich "don't pay their fair share" when the rich are the only ones actually paying income taxes, I would say your argument has already been disproven.
I still think the kernel of the idea is a really interesting one. Could you make it work, by, say, ranked-choice voting to allow trading off conflicting priorities? (You wouldn't even need to get people to explicitly rank them, come to think of it - you could just prioritise the more popular 'yes' questions.)
I vote green/left, although they tax me higher, because I think climate change and social justice is more important than whether or not I have 1k per year more or less.