Why are we assuming the "budget stalemate in the US Congress" is a problem that needs solving?
For me, it's because I have a basic grasp of politics and the economy.
The founders, for obvious and excellent reasons, tried to structure the system so that violently controversial stuff could not get done, so that to govern effectively you needed a widespread consensus.
There is no consensus on spending unprecedented gobs of money, and it is not obvious that there should be such a consensus. Hence, gridlock. It is not a bug, it is a feature.
If the only effect of gridlock were to limit spending, that could be a feature. But the effect of gridlock in this case would be to stop government spending suddenly, and put a few million people out of a job suddenly, as well as immediately downgrading American bonds and the dollar, and other consequences as well. Calling this a feature is like building a train that jumps off the tracks if it goes over 100MPH and calling it a safety feature.
The budget stalemate in the US Congress was caused entirely by blocks of voters and representatives that coalesced around strong sets of opinions that few people would have come up with on their own, and by political party leaders forcing representatives in their parties to toe the party line. Politics isn't the mind killer. Political parties are the mind-killer.
Parties are also notorious for obliterating information in elections, as well as for encouraging voters to vote sans information. If you went to your polling place and saw a list of candidates, none of whom you'd heard of before, you might rightly refrain from voting and polluting the signal with your noise. Knowing party affiliations makes people think they have enough information to vote.
For discussion:
We want the freedom to form groups that promote political concerns. But it would be possible to keep these groups at a greater distance from elected representatives. Candidates for office could be forbidden from endorsing a particular party. The Congress could be forbidden from basing any procedural rules on party affiliation. Political parties could be forbidden from making large donations to election campaigns, or sponsoring advertising. That's not so different from what we do today with religious groups, which are not much different from political parties.
Political parties are currently officially part of Congress' operation, even though they're not in the constitution. There are all sorts of Congressional rules specifying how the parties interact, who gets to choose committee members, who runs the House and Senate floors, etc. A party leader can punish a representative who doesn't toe the line with many incentives and disincentives.
Make that illegal. Make persecuting a representative for party-based reasons have the same legal standing as persecuting a representative for religious reasons.
I will ignore comments saying "you're an intellectual dreamer", for the usual reasons.