“[S]ubject to the jurisdiction thereof” would not be necessary had the framers of the amendment intended to grant birthright citizenship. Had they intended that, they could simply have written that all persons born in the United States are citizens of the United States. That they did not tells us something.
The phrase “subject to the jurisdiction thereof” means “owing allegiance thereto” — which illegal aliens don’t. Their presence is by definition outside our laws and they still owe and retain allegiance to whatever rotten little satrapy they came from.
So of course they can be denied birthright citizenship without amending the Constitution. And they should be.
There's no need or justification to make the words into something else. It means exactly what it says, "subject to the jurisdiction". People that are subject to the law, and not people who are not.