You made the most crucial point on immigration. Before everyone turns themselves into a pretzel about deportation, first priority must be to close the borders.
When you boat is taking on water, first priority is to plug the hole. Only then you can begin paling the water out.
Illegals who arrived as tourist or students and overstayed their visa’s is entirely another matter. When I was on a student visa, I was required to report my address to immigration every 6 months. That was early 1960’s. Do not know what current requirements are.
The law hasn't changed, but enforcement has. Universities are not cooperating with the Federal government and don't report the information without consequence.