If I go into your house with a gun in my hand and force you to give me all your money, but, being the good person that I am, allowing you to chose whether I would spend that money on an extremely poor school I force your kids to attend, or on a brutal police force, that treats you like shit (and, how practical, continues this type of theft for me in the future), or on welfare programs, that do nothing but to make the poor even poorer, would you seriously consider that a function of the free market? That's just insane.
If you are assuming that by this technique the least bad branch of government is going to be relatively better funded than the worst branch, than this might actually be true (assuming for the sake of the argument that all taxpayers have similar preferences on what's good and what's bad). But this doesn't explain why having taxation in the first place would be better than having no taxation at all.
Also, with no government at all, or if you are a minarchist, with just as much government needed to guarantee for a functioning market place, why couldn't people freely fund organizations their products they like even better than under your desired system? What does the concept of taking some of their money coercively add? Also, how would you decide how much one must pay and what the available options are at all?
And in regards to the "sound money" discussion, I guess you don't get the point of it. If there's still a central bank and legal tender laws, etc., what stops any branch of government that loses funds under your system (because tax payers aren't satisfied any longer) from simply issuing bonds, knowing they get bought up by their friends at the central bank at the cost of all the "money users" (inflation)?