".........regularize everyone on the US side of it."
Wrong.
No Shamnesty.
Fine. Don't give amnesty.
Close the border, and let the 20 million illegals this side of it continue to work and hide as best they can. Deport the ones you catch. The problem still takes care of itself in a generation.
Closing the border is a good idea to manage the situation.
Taking away citizenship by ending the 14th Amendment is a bad way.