Are illegal babies -- "subject to the jurisdiction thereof"? -- I'd say not, -- if Congress properly enforces "the article by appropriate legislation".
Anybody inside the country is subject to the jurisdiction thereof. They may not have full citizenship rights, but they have some constitutional protection. The Fourteenth Amendment was a major revision of the structure of the state and enhanced the federalness following the Civil War and which enhancement continues to this day. One of the writers of the Fourteenth Amendment said it allowed for the corporation interpretation and said it was specifically discussed although there is nothing to that effect in the minutes.