I think we need a figurehead in the US, like your queen. We could elect our own king or queen every 4 years, who could be a ridiculous celebrity: he or she could live in the white house, give dinners, tweet crap for us to talk about, say dumb things for us to make fun of, and provide us with all the celebrity scandal and gossip we obviously crave.
Meanwhile we could have serious, hard-working politicians who could deal with international relations, racism, economic inequality, health care, education, and most of all, the terrible environmental crisis.
I think that since the US has such an impact on the rest of the world, any citizen of any country we are allied to should get a vote in the serious election, if they would like one. Perhaps to be eligible for this vote the non-US person should have to take a short test, which would consist of pictures that the person has to look at without feeling uncomfortable. I would have pictures of a gay couple with kids, a woman driving a truck in a short skirt, a man wearing an apron and changing a nappy, etc. To get a vote, a person would have to nod pleasantly at each of these images.