Certain US traditions would make it culturally difficult to install a monarch, but I'm sure a relatively powerless figurehead president would do just as well--many parliamentary republics have them.

We could adopt your system but add a head of state called the President, available for ribbon-cuttings, photo-ops at schools and state visits, who could be popularly elected, retaining the four-year pageant but removing most of the unpleasantness. Then the House could elect someone to head the executive departments... hmm, "prime minister" sounds foreign; let's call that person the Chief Secretary. That sounds about right.