Jefferson invented two-party politics to counter a growing threat of tyranny from the one-party rule favored by the Federalists. The alternative is not partyless politics, it is either return to that single party rule or government by multiparty coalitions of the sort you see in parliamentary systems. In the US structure with its independent process for choosing the executive leader, it would necessarily be the former of those two.
The problem with US parties today is an overconcentration of control in the political class. That has taken root in the two major parties as well as in the government overall. Both the demwits and the retardicans now represent the national status quo, and their party establishments along with an allied media control politics. The only genuine grass roots movement we have seen in a long time is the TEA party, which arose spontaneously from an outraged public as Washington showed its true colors once one-party rule was reestablished. That reform movement has now been subverted, institutionalized and coopted. Doing so was the only bipartisan work done during the last three years. The demwits' role was to demonize it to destroy its effectiveness, the retardicans' was to draw it into their fold. They succeeded, and the TEA Party is no longer a threat to the status quo.
US politics without the two-party system would be similar to what we saw during the first two years of the Obama administration, but more so. It would be rule by one party, unrestrained by political competition and, since the Constitution is now dead as the men who wrote it, unlimited by legal boundaries. It would be totalitarianism. No, thank you. Bad as it can be, the two-party system is far better than that.