answer WHY the Founders found it necessary to distinguish between natural born Citizens and just anyone born here?
The Founders of 1776 are not relevant. The Framers of 1787 found it convenient to write a Constitution and consolidate the existing 13 rules of naturalization to one Federal rule. As they were creating a much more powerful central government, they found it convenient to protect against alien born persons attaining control of the U.S. government or armed forces.
The Hamilton Plan phrased it in the simple English of a non-lawyer. "No person shall be eligible to the office of President of the United States unless he be now a Citizen of one of the States, or hereafter be born a Citizen of the United States. The lawyers used the legal term of art from the English law, natural born. A natural born subject or citizen is one who becomes a subject or citizen at birth, as opposed to any time after birth.
They did not, as you say, distinguish between natural born citizens and just anyone born here. They distinguished between those born aliens and those born citizens. The only ones born here and not born citizens are those born to a parent who enjoys immunity from our laws, such as an accredited diplomat. In that case the child has derivitive immunity and is considered to have been born in the sovereign territory of the parent.
Naturalization is a legal process for aliens that occurs only at some time subsequent to birth. Natural born citizens are those who acquire citizenship at birth due to the circumstances of their birth.
Founders/Framers, okay pedant, totally different people right?
“They did not, as you say, distinguish between natural born citizens and just anyone born here.”
The hell they didn’t! They SPECIFICALLY wrote into the Constitution the requirement for President “Natural born Citizen”.
None of your ‘Hamilton Plan’ obscuration bs is in the Constitution is it?
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.