One system that I've been pondering lately (maybe it has an official name), is that you directly choose where a portion of your tax dollars go. So on your tax return, 50% (just pulling a random number here) automatically goes to the government as it does now, to spend as they see fit. The other 50% is yours to spend, so to speak, within the constraints of maybe a few dozen pre-selected options. If you want it all to go to the military, then that's your call. Some to infrastructure, and some to healthcare, then, boom, done. In an ideal system, you'd have 100% control, but you can only trust the public so far.
Which brings up the point of whether one of the basic theories of "pure" democracy is sound: that the public can rationally govern itself. I don't think current democracies really prove it one way or another, but if the theory doesn't hold water, then my idea would probably be a disaster. Fine-tuning the percentages could yield some interesting results though, effectively reducing the importance of who gets elected in the first place.
What you're proposing is a specific implementation of direct democracy. It sounds nice, but it's fraught with the very same problems that representative elections face; every tax option would have its own campaigns with distortion-filled ads, people would likely vote to funnel vast amounts of money toward things that actually don't need it based on emotionally manipulative ads they saw... I'd rather try to fix representative democracy than experiment with direct democracy.
I don't see it as necessarily replacing representative democracy, but complimenting it, giving people more of a say in how things are done on top of traditional voting. If I could explicitly choose how even 5% of my taxes were spent, that would give me a very tangible feeling of influence that I just don't get from voting, at least at the higher levels of government (as an american). It would also push me to do my homework about issues a lot more, but that might not translate to the public at large.
Agreed about the potential for the system to be hacked by powerful entities though, perhaps even worse than it is now...tough to get around that one.
The closest you have to that is making charity donations and offsetting that against tax. In that case you are diverting money away from central government and putting it elsewhere.
Many government projects require a certain level of investment over a number of years or even decades. It wouldn't be efficient to start say a fighter jet development program if you might have to ditch it half way through because public opinion changed.
True, it wouldn't apply well in all situations, but if you tweak the percentages correctly, only giving the people X% of control over their taxes (maybe relative to some metric like income or education), then maybe the government could have enough of a buffer to protect against swings in public opinion for sensitive or critical projects.
But your point about the fighter jet program could actually be an argument for this type of direct democracy. Let's say the public found out that the program was causing, or somehow will cause, the deaths of thousands of innocent people in a third world country. The military could then face spending cuts, directly dictated by the people, that would threaten the program, and rightly so.
That would last until the first terrorist attack or economic crisis and then tend towards 0, because "we really need this money".
Besides you would probably be able to work around this with creative accounting. Let's say everyone wants to defund the military and put more funds into education; military training establishments now show up on the budget as "defence studies schools" because they are issuing general educational certificates to soldiers.
I think fundamentally if you think some particular government is going to just waste large amounts of tax money you should really just not vote for them. You vote in a government because you want them to make sensible choices in how to allocate resources for public projects so you can concentrate on your startup or whatever.
Sometimes this stuff can work on a local level. For example schools can have governing bodies of volunteers (usually parents) who get some say over how the school budget is used.
You might be interested to know that there's a voting system for politicians which works similarly, called "Mixed-member proportional representation." 50% of the politicians are directly voted and then 50% are appointed in order to make the party distribution look more like the voters should have expected it to look. In the analogy you're proposing, "parties" become the "bureaus" of the bureaucracy. You can also imagine their "programs" becoming a sort of "party list", I suppose, and you could even do a party-list proportional election in that sense. (Imagine if the 50% was instead 100%.)
The problem is that "fixed seats" are the wrong way to look at this; the fixed quantity is actually the budget money made available by the government. So, the last budget was a $3.54 trillion budget, and we might be able to "chunkify" the budget into $20 billion "awards" and thus "elect" 177 programs in the various departments and agencies to be allocated those awards. The US has 15 executive departments which is about the limit which democratic elections on this sort of scale usually take, so that would probably be the limit. Each department would put several large "programs" in order and you could choose some programs and some departments, or so; we then fund the most popular programs, followed by the top programs in the funded departments, until the department allocation matches the national average.
The problem here is: with political parties, anybody can start one. But the executive branch starts new executive departments and presumably builds those prioritized lists. Suppose the executive branch decides that there is only one department, they could more or less set the entire budget as they pleased.
Which brings up the point of whether one of the basic theories of "pure" democracy is sound: that the public can rationally govern itself. I don't think current democracies really prove it one way or another, but if the theory doesn't hold water, then my idea would probably be a disaster. Fine-tuning the percentages could yield some interesting results though, effectively reducing the importance of who gets elected in the first place.