Seeking the most vicious common denominator
Certainly it's very idealistic to suppose that we could ever get along without political parties, or organized factions within those parties. The enormous institutional inertia that keeps our elections, for the most part, a two-party race might help to restrain divisive factional energy somewhat. A lot of the more unpleasant factional battles take place intra-party, and tend to get resolved during primary season at the latest. In systems where lots of narrowly focused parties band together to form coalitions, the constant level of political discord seems a bit higher.
I would imagine voters in those systems feel a little more intellectually honest during elections - they get to vote for parties that more precisely align with their political beliefs. But then they get to watch their elected representatives make deals that violate those principles, in order to form governing coalitions. If someone like Ron Paul ever got elected President here, his energetic followers would have been very disappointed by the post-election compromises he had to make.
But the real problem I perceive with the divisiveness of our political system is the way centralized Big Government plugs all the healthy escape valves for political tension. When power is decentralized, people have a decent chance of banding together with like-minded folks to affect local laws; those who don't like the way things are going can move somewhere else. Trapping everyone together beneath a huge, invasive single government squeezes us all into a pressure cooker, and makes membership in bitterly adversarial political collectives more important. The battles between those collectives are zero-sum games, in which victory for some requires defeat for others. When the creators of ObamaCare "won," everyone who disagrees with them lost, and there is no way to escape and build a competing system that might win broad popular support by succeeding where ObamaCare fails.
The absence of competition between ideas increases factional bitterness, because you can't try an approach that differs from the prevailing federal orthodoxy and demonstrate that it works better. Competition is inherently persuasive - people voluntarily gravitate to the stronger competitors. Political control, on the other hand, is coercive. We have an election, the winners set about "transforming" society in various ways, and dissenters are angrily informed that going "backwards" is impossible. No wonder every faction in our great republic is playing for keeps. The political class keeps assuring us that defeat is permanent.