You are on the correct track, but that statement, as worded, may not be adequate to cover peripheral circumstances.
Example: A couple comes to the USA legally as immigrants and they have a child before they finalize their citizenship. Does that make the child a citizen immediately? As it stands Now, the child of parents born outside of the USA is a citizen even if only one parent is an American Citizen.
That would likely be a subject detailed in peripheral legislation but as an Amendment I'd recommend putting it up for debate. Children are the responsibility of their parents, not the Government, and if illegals are removed
for reason the child should go with them and should no longer be considered a citizen.
The Amendment process is long and difficult for good reason but I like your idea.