A simple majority is 50% plus 1. In order to achieve it both sides (all sides) will seek a coalition with all who will vote with them. This type of “inclusiveness” requires the majority of each party, who I believe are mostly centrist or moderate, to ally themselves with those at the extremes and to play to them. The extremists must get something for their votes, so the entire party shifts towards one extreme and the different parties become more opposed.
I suggest that probably two-thirds of both parties in the USA today are moderate folks. If so, that means that two-thirds of the nation pretty much agree on most things. It also means that we are letting the one-sixth closer to the far left and the one-sixth closer to the far right tear the nation apart.
We see in the Senate today a two thirds rule on the question of impeachment. But what that two-thirds rule means is that, in an evenly divided body, only one-sixth of one party must agree with the other side to settle the question and that means that as much as two-thirds of one party will not be in accord with the settlement. Even this two-thirds rule perpetuates division.
One way which might eliminate most of the divide would be to require a 50 % plus 1 vote from both parties on all matters of substance. With this way we would be assured that more than half of all the nation would be in agreement. We need to do something to force ourselves to become unified. The way we are going now is not sustainable.