Yes, of course they would care.
Why did the founders add the natural born citizen requirement only for presidential candidates? That question is well worth the time it takes to discover the answer.
Representative - No Person shall be a Representative who shall not have attained to the Age of twenty five Years, and been seven Years a Citizen of the United States
Senator - No Person shall be a Senator who shall not have attained to the Age of thirty Years, and been nine Years a Citizen of the United States,
President - No Person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President; neither shall any Person be eligible to that Office who shall not have attained to the Age of thirty five Years, and been fourteen Years a Resident within the United States.
Natural born doesn’t refer to the candidate’s place of birth; natural born refers to the place of birth of both of the candidate’s parents.
Right, but I think it's whether his parents are U.S. citizens not his parents place of birth.
"Natural born" simply means citizen by birth, as opposed to citizen by naturalization.
There is more than one way to be a citizen by birth. All born in the US — except those not subject to US jurisdiction, a very narrow class — are citizens by birth and, therefore, natural born. Additionally, all born abroad to one or more US citizen parents are citizens by birth and natural born.