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The public could vote with their taxes what programs get funded.
Voluntary participation is a good way to source genuine motivation.
Maybe only to certain degrees. Those that pay more taxes may have too much influence.
A certain percentage could be allocated with a certain percentage going to critical programs .
What OP described is actually a free market transaction. Which does very well at allocating resources and is the closest you'll ever get to "the will of the people" whatever that means.
The British government does this with small portions of tax income. On certain collection forms you can put essentially "I'd like my five quid to go to __". This resulted in a net increase in people paying taxes.
We already do, in a way, through elected officials. That's what a representative democracy is about, isn't it? A direct democracy would mean each person could specifically choose where their taxes go, but it is hard to implement and maintain and might result in richer people having more say.
Knowing where tax should go is complex, there is a risk for tax to be badly allocated.
If It was up to society the allocation would be simply wrong. E.g.: Nobody would pay for politicians who get their salary from taxes; or for military; or probably would overpay shelters. So, lazy people will get lazier and strong would loose motivation. That's socialism, lol
Then it would be charity, not taxes.