The key, as you pointed out, as did Buttigieg, is to identify areas where there are shared problems you can work together to solve. There's something about local government that seems to encourage this, in my opinion. Maybe you think gay people are awful and I go the pride march every year. Guess what, the water main is still leaking, and building permits take too long. Maybe we can just focus on working together to solve real problems, and accept that there are a range of social issues where we probably won't ever agree? And that's just fine?
I think we used to know how to do this in the United States, but something changed.