You left out a clause. They not only need to be born here but they also need to be "subject to the jurisdiction of". Now we have repeatedly argued in this forum about what that particular clause means so I am not going to engage in that discussion again but the bottom line is that it will mean whatever the Congress and the Courts decide it means. The Constitution gives solely to Congress the power to establish our immigration policy. The Constitution also gives to Congress the power to limit the jurisdiction of the Courts including the Supreme Court. It seems to me that if Congress has enough desire, they can limit birthright citizenship with action that falls short of a Constitutional Amendment.
jurisdiction n the power, right, or authority to interpret and apply the law : the limits or territory within which authority may be exercised
Illegal parents, I would expect, can be shown to not be "subject to the jurisdiction thereof". Children are subject to their parents; would this also place them outside of the "jurisdiction thereof"?
It seems to me that you are right, so why has congress failed to exercize its authority?
I have felt for a long time now that congress's true will is carried out by the courts. They let the courts do what they know their constituents would not approve of. The why of it is what alludes me.