Not exactly.
First, I don't think the Framers thought they WERE allowing "persons with a foreign allegiance" into the White House.
I simply don't see any indication that they saw persons born on US soil, though their parents might not have naturalized, as having "a foreign allegiance."
Part of this may be due to the different world they lived in. The odds of a person born here going to the country of his or her parents' origin were virtually nil. That's at least a little bit different now.
Still, I KNOW people born here with immigrant parents. The kids grow up speaking English. And if they ever go to visit the parents' country of origin, even if they speak the language (and quite a few don't), they understand very quickly that that country is NOT "their" country.
And that's TODAY.
Secondly, I think that if the Constitution were worded the way you demand, that James Madison, at least, would feel that gave it a "tincture of illiberality."
And I base that statement on Madison's own words.
Third, even if there were a potential benefit to establishing a rule, it doesn't follow that the Founders MUST have established every rule that could POSSIBLY have a potential benefit.
I would say there would be a potential benefit to Constitutionally disallowing all persons with a felony conviction from being President. But the Founders didn't forbid that, either.
“Secondly, I think that if the Constitution were worded the way you demand”
That is the single stupidest comment on the thread. What is it w you people—i.e.: the people who advocate for POTUSs w foreign allegiances/divided loyalty? Are you really this thick, or do you misunderstand on purpose???