Yes, that is an exact and concise statement of what the Trump administration will argue.
Most, if not all, states have laws that require all parents, including those here illegally, to send their children to school. If they don't, the government can force the children to attend school. Both the parent and the child can face legal consequences for truancy. The children are subject to the jurisdiction of the U.S. That is true regardless of whether or not the child is born in the U.S. Being born in the U.S., though, affects citizenship.
A diplomat, on the other hand, is a different matter. As we all know from Law and Order, diplomats and their children are not subject to U.S. criminal laws. If a diplomat or the diplomat's child commits a crime in the U.S., they will have to be prosecuted, if at all, in their home country.
Trump violated his oath -- on the same day he swore it -- by signing an executive order that he knew was unconstitutional. A normal person who did that would feel shame.