Or just follow what one of the authors of the Amendment put in there when he wrote it ...
Jurisdiction understood as allegiance, Senator Howard explained, excludes not only Indians but “persons born in the United States who are foreigners, aliens, [or] who belong to the families of ambassadors or foreign ministers.”
Furthermore, there has never been an explicit holding by the Supreme Court that the children of illegal aliens are automatically accorded birthright citizenship. In the case of Wong Kim Ark (1898) the Court ruled that a child born in the U.S. of legal aliens was entitled to “birthright citizenship” under the 14th Amendment. This was a 5–4 opinion which provoked the dissent of Chief Justice Melville Fuller, who argued that, contrary to the reasoning of the majority’s holding, the 14th Amendment did not in fact adopt the common-law understanding of birthright citizenship.
What you say is accurate. However the expanded view of birthright citizenship based on Ark V. US has been the guiding principle of law since that case was decided. To change the principle of birthright citizenship will require either the Supreme Court to change it’s opinion in some future case, an amendment altering the language of the XIV Amendment or cancelling the XVI Amendment all together. To end birthright citizenship will require one of these actions to be successful.