I’m not sure what you propose works under a republic. You’d have to forswear allegience to the Queen again to get a comparable situation.
The problem is, as you’ve noted, is that anyone could become president once they were naturalized. How much sense does it make to reject foreign entanglements, when someone could come from abroad and govern the nation?
Granted, you’ve got the worst of it right now with Obama, where someone has done just that, come from abroad to govern America, *despite* the laws in place that say this shouldn’t happen.
This is a disaster long term for the republic.
Anyways, long term, that is one of the benefits of the monarchy to retain the general character of the nation. I’d accept your argument if it meant going back to how things were done prior to the revolution, but I sincerely doubt that many Americans, even those who are sympathetic, would support this. We’re both odd-ducks.