Theoretically, could we balance the budget painlessly?

Improper aggregation. Neither consumption nor taxes nor government spending is amorphous and evenly distributed.  Different mechanisms have very different effects in WHO is able to consume more or less, over what timeframes.  

Note that most government spending is eventually paid to workers and contractors, who use it for consumption, so it's not clear that in aggregate it matters at all.  It's definitely clear in detail that it changes who consumes and when the consumption happens.

A good thought experiment is to compare/contrast debt reduction via inflation (just print enough money to pay it off) vs via income tax increase (non-targeted; just double the values that hold today) vs via land sales (the US government has a LOT of valuable property that could be privatized).  Your analysis of and reaction to the impacts of these things will tell you a lot about your biases/priors about how wealth works in modern economies.

edit: another way of thinking about this (but reinforcing that money only matters when it's not evenly distributed) is to realize that debt is a BALANCED timeshift.  For every seller, there is a buyer.  Government (and it's beneficiaries, employees, contractors, etc.) get a bit more consumption now, with a pretty nebulous reduction in consumption later (for a different subset), and lenders (buyers of bonds) get reduced consumption now, with a well-defined additional consumption later.

That disconnect in the otherwise-balanced equation (lenders know when and how they get paid, borrowers don't like to think about it and mostly believe other subsets will get taxed more than they) is absolutely core to the belief that government solves problems.

"Could this work?" and "Could we get there from here?" seem like separate questions in this case.

Could it work in general, starting from a clean slate? Probably. In a tight feedback loop between personal taxation and government spending, voter demand would probably tune government spending so taxes stayed at a tolerable level to allow an adequately satisfying amount of personal consumption.

Set the goal as "do not increase the current deficit", and the link between how much the government spends and how much the individual pays could be concrete enough to be comprehensible to even the below-average half of taxpayers and voters, with a good enough publicity campaign. Hypothetically, a tax return could say exactly how many dollars of your payment went to which initiatives in this system, and with sufficiently legible returns correlated to electoral platforms and feedback on elected politician voting/performance, voters and taxpayers could experience the sensation that they have some say in where their money goes.

Could we get rid of the existing deficit this way? I doubt it. American cultural values say it's bad to punish someone for the actions of others, even if the others are related to them in some way. That value is why almost all personal debts stop at the estate, and aren't inherited by the heirs. Demanding that current taxpayers foot the multi-billion bill for money spent by their ancestors directly violates that value. Although you can ease the populace into tolerating all sorts of horrible things in a frog-boiling sort of way (look at the expense of filing one's taxes at all, perpetuated by tax preparation lobbyists), sudden widespread changes tend to be much easier for the populace to comprehend and resist.