When they could get anything done they delegated a lot of power to the Executive. Which worked ok, but eventually a "unitary executive" appropriated even more power, and the Legislature is powerless to prevent it.
My third (previously second) is outlawing political parties. The problem with that one is it would be really difficult to implement in a way that doesn't run afoul of freedom of association and freedom of speech. Probably worth figuring out though.
We are too gregarious to prevent emergence of political groups. A parliament of cats would probably be more individualistic, but not that of humans.
Egypt after ousting Mubarak held an election where a third of seats were reserved for independents. Most winning candidates were just Muslim Brotherhood affiliated. I suspect the military interim government did that deliberately to justify their later coup.
This is where the intra-party coalitions become important. Every party of significant size has them. Labour is effectively a coalition between a rightwing faction (New Labour/Blue Labour) and everyone else who is more leftwing. The internal and external debate is the question: should they focus "right" (immigrant and queerbashing, welfare cuts) to appease the right wing of the party and try to pick up Reform/Conservative voters, or focus "left" on their base and people who are switching to Green?
My personal opinion is that Mitch McConnell's intransigence and unwillingness to do anything lest Obama get credit for it led directly to an increased desire for a "strongman"