The Southern and Northern Methodists, Dutch Reformed, Church of Christ and several others have declared themselves to be united in purpose. Hence the name "UNITED Methodists.
The Roman Catholics, Episcopals, Anglicans, Eastern Orthodox, Syrians and several others recognize one another's baptisms marriages and such.
The proliferation of sects in the US mostly is the result of Southern Baptists (who do not have any sort of strong hierarchy) and the fundamentalists, who are often organized around a single preacher in some sort of mega church.
You are correct, however, in saying that it is very unlikely that all the Christian churches will all ever unite.
I recall a conversation with a woman who said "We ought to have prayer in the schools: there wasn't so much crime when we had prayer in the schools."
I asked her to suggest a prayer that everyone could agree upon.
She said "I don't want to talk about this any more."
Of course, the per capita crime rate was higher then than it was at that time.
But, of course, there was a LOT MORE crime being committed on teevee.
She could have solved this crime by just getting people to use different toothpaste or detergents.
