Your idea is a disaster.
First off, defence and the safety net account for the lion’s share of the budget, so any changes in allocation of the remainder wouldn’t make that much of a difference. If I was in charge of determining the budget, I’d absolutely make massive slashes to defence spendng, and probably make some minor trimmings to health care spending, so I’m not especially impressed by the option to choose where the rest of my taxes go.
Secondly, most people aren’t particularly good judges of how resources should best be allocated: some of them are simply ignorant of the details of running the government, and a lot of us are blinded by economic self interest.
Thirdly, I don’t think you’d get agreement on ‘what areas of the budget are too important to be voted on’. Person A might argue that funding for crisis pregnancy centers is sacrosanct. Person B might argue the same about wildlife conservation efforts. Person C might argue that about alternative energy funding. Person D might say it about relief aid to Africa. Personally, I think all four of those are better uses of US government funding than funnelling money to ‘democracy’ activists in the Ukraine, Venezuela, and Thailand, but that’s just me. In any case, you aren’t going to get agreement on what those top prioirities are, so the whole thing is dead in the water.