The reading I've done has folks coming down on both sides of that. Some concluding that that Congress can (your opinion) and other that it can't override by statute what was done by this phrase in this constitutional amendment.
Popular opinion seems to desire opening the office of president up to all citizens, regardless of they come to that status. Again, my impression is that nationalism is destined for the trash heap, and we're more likely to see the office opened up too all citizens, than we are to see it limited to children born of citizens.
-- It's less a matter of 'was it intended?' and more a matter of 'what does the law say?' --
Seeing as how this is constitutional interpretation, the rhetoric of the founders (what does it say), plus the meaning of certain terms of art at the time and any record of rationale for the choice of certain words and phrases (what was intended), will all play into applying the words of the constitution to a fact pattern.
I speculate that the drafters of the 14th may have been focused on issues other than opening the door wider for the presidency, hence they did not make perfectly clear how "natural born citizen" was to be applied. While the base constitution uses both "citizen" and "natural born citizen," the 14th amendment does not expressly confer "natural born citizenship" to all persons born in the country. It could have, just as the Wong Kim Ark Court could have said that Wong Kim Ark was a natural born citizen.
And too, should a federal court, ultimately appealed to SCOTUS, decide to clarify the point, the holding of Wong Kim Ark may be extended to say just that. It's a very short clarifying jump; although IMO it is contrary to the principle of reserving the office of president to a person with undivided loyalty. If my father was the King of Saud, and I was both the prince, and the president of the United States, well, you get the picture.
“The reading I’ve done has folks coming down on both sides of that. Some concluding that that Congress can (your opinion) and other that it can’t override by statute what was done by this phrase in this constitutional amendment.”
True. I’m all for Congress at least trying. WOnt ever happen until we get a GOP majority back though.