That seems like a very good demonstration of the pitfalls of originalist interpretations of the Constitution. Even then, the argument comes off as extremely weak. And it doesn't even begin to try and address the consequences of reinterpreting the meaning of "and subject to the jurisdiction thereof".
Are conservatives envisioning a new class of slaves? People born on US soil who have none of the protections of the Constitution? Even if that is not the goal, it's not hard to imagine that there would be far-reaching consequences from deciding that the Constitution was not a limit on the behavior of government, but in fact only applied to citizens. What a massive bump in power for the bureaucrats in DC.
Heck, we could just snatch people off the street and declare they cannot prove they are a citizen therefore they have no Constitutional protections. No right to due process so they can prove they're a citizen, nothing like that. Better plan on carrying your passport at all times (and hope it doesn't get ... lost).