Re: Glad they didn't overthink the process....
The Presidential model follows the goals of a Monarchy in a functioning democracy, but rejects the idea of hereditary rule. Why do you think that's "far worse"?
The US has Trump for 4 years, (and though it pains me to say it), possibly 8. Big Liz the Deuce has been God-King-and-slaveowner for the better part of seven decades. At least her powers are largely ceremonial, but that's only a "Gentleman's agreement": a future monarch could quite easily restructure Parliament and the Courts on a whim, or dismiss them entirely.
In the US, allegiance is sworn to the Flag of the US. In the UK, it's to the Monarch.