This sounds similar to the case of Obama. His father was not a citizen and his mother was. At the time I believe there were statutes which covered whether or not Obama's mother was able to pass on citizenship. THAT is not "natural born".
The status of Awlaki as a student would not in any way diminish his loyalty to a foreign power nor would it eliminate the conflict of loyalties in his children.
I would say that such a child would NOT be eligible.
And you're hearing such an opinion from one who has two grandchildren born of a foreign national. I am convinced that neither of my grandchildren should be eligible. After Obama, however, anything goes.
If you recall, Awlaki was the US CITIZEN authorized by Obama to be killed.
So, his child was a child of a US citizen born while in the USA, but now living the Lord knows where in Yemen probably.
So, do you have problems with this child being eligible to the presidency. It was BORN IN THE USA to a US citizen.
The problem with Obama’s mama was that even though she was a US citizen, if he’d been born overseas, she was young enough that she wouldn’t have met the residency requirement. This is quite different than Cruz whose mother more than met the residency requirement.