The issue wasn't alien "born". The issue was alien loyalty.
Hamilton was Alien born, so was Thomas Paine, and so were many of the early Framers/Founders, but the issue they were concerned about was that of loyalty and allegiance.
None of that is solved with the British derived definition of "citizen", but it is all completely resolved with the Vattel definition of "citizen."
The fact that we use the word "citizen" (uncommon in the English of that era) is a good indication that the intent always was to follow Vattel.
Had they intended to follow English law, they would have kept "subject", which was the usual and common way of describing a member of a nation at the time.
"Citizen" was adopting the Swiss usage, not the English usage.
Chester Arthur.