Article II specifies citizen at the time of adoption of the constitution, and natural born citizen thereafter.
Citizen necessarily encompasses naturalized citizen as well as natural born citizen. If a naturalization statute creates natural born citizens then there is no need for the Grandfather Clause, citizen would have sufficed.
Equating "born a citizen" with natural born citizen is an impermissible construction.
No, there's a huge difference between someone who is a citizen at and by birth, and someone who was NOT born a citizen of the United States but was naturalized at some point later in life.
Those who are citizens AT AND BY BIRTH (whether that status can be affected by laws set by Congress or not) is a natural born citizen. Those who aren't citizens AT AND BY BIRTH are not.
See my previous post. James Bayard wrote in 1834 that people in Cruz's situation were ELIGIBLE. And Chief Justice John Marshall, who of all persons ought to have been in a position to know, agreed.