Not totally absurd
I don't think the logic being used by the right wing will hold up here in court, but it isn't totally absurd. The "subject to the jurisdiction thereof" clause in the admendment seems to have been added specifically to address related problems. Children born to diplomats for instance, are born on American soil but not subject to American jurisdiction.
American Indians where not all American citizens until Congress passed a seperate law making them such under the same principle. More directly in this case, a foreign power that invades and occupies American soil would eventual result in children born to the invaders who can not claim American citizenship. In principle anyway, I'm not aware of any case of that last part being applied in practice.
However, I don't think any court will actually accept an argument based on that. To accept it would raise the question of the US having jurisdiction over illegal immigrants in general. And even given the courts current extreme deference to Congress, that would be too fundamental a contradiction to accept.
Having though about it for a while and looked at the statistics, I don't think this is a big enough problem to do anything about. Most of the children born to illegal immigrants are born to couples that have been in the US for some time. The group that intentionally cross the border to have their child in the US in very small.
As a result any 'solution' would have worse consequences down the road. A narrow rule that let INS deport children if the parents specifically entered the country illegally to have their child here would create another messy legal tangle that would create as many problems at it would solve. A broad rule would create a perpetual non-citizen underclass, with so many bad consequences it becomes hard to list them all.
Jay
