There is no way that our founders could forsee the problem of illegal entry on the part of millions of non-citizens. However, to repeal the 14th amendment and replace it with a new amendment is an exercise in futility because our present day political structure would kill the process in a thousand ways.
What could be easier without changing the amendment is to drive an interpretative decision through the courts based on the meaning of:
... and subject to the jurisdiction thereof, ...
Since illegal foreigners and their children are not subject to the jurisdiction of the United States, they should be deported back to the countries that do have jurisdiction.
Now legal foreigners who have a valid visa or permanent residence are subject to the laws and jurisdiction of the United States. They must have agreed to this prior to accepting their visa or residence status.
The continual mantra of the Living Document Grenadiers.
The same arguments apply to carrying Assault Weapons, tapping telephones, requiring ID Tattoos, etc.