Senator Cruz did not have to be naturalized.
But what of James McClure? You yourself pointed out that John Armstrong described him as a "naturalized" citizen. John Armstrong was well aware that he was BORN in Charleston because McClure had the papers to prove it.
The fact that Armstrong referred to him as "naturalized" (and John Armstrong was in fact a Delegate of the US Congress in 1787 and therefore ought to know what was the common understanding of the time.) indicates that the United States did not follow the Jus Soli rule for citizenship, but instead explicitly followed the Partus Sequitur Patrem rule.
I think much of the common understanding of James McClure's citizenship status has remained undiscussed. We have a dichotomy here. We have a condition in which the facts do not comport with what is alleged to be the common understanding of the time.
Armstrong, in a confidential memorandum to James Madison, described James McClure as a "naturalized" citizen. How do you respond to this fact, seemingly in conflict with the argument you would have us believe?
Because like expatriation there was not unanimity of thought on citizenship. That’s why you can find different people saying things that are contradictory (Judge Roberts and William Rawle or James Madison and James Jackson).
As you know McClure was given a passport by the American Minister in London that declared him to be a native American. You say it might have been a forgery, but why would he need a forged passport when he could get one under two different theories. Mr. John Rodman for example believed him to being a citizen based on his place of birth. And we know that when Secretary of State Monroe wrote a letter on McClure’s behave he never mentions the father and only says he was born after the Revolution in South Carolina.