That is a false premise. It may be what happens, but it is not required by any means. All that is required to make illegal aliens a tiny presence in our midst is to enforce existing law. That means employer sanctions, real big fines and prison. That means roundups. Start at big employers, that would make a big splash. Plus, you can fine the company millions, which would pay for the expense of the operations.
Enforce SS laws and imprison those with false papers, felons. Pass laws that take away automatic birthright citizenship, schooling, free health care except in emergencies, welfare.
Deputize local law enforcement. When they are stopped for a traffic ticket, it will end in deportation.
If they can't drive, and can't have a job, millions will go home. Stragglers will get picked up here and there. When benefits are lost, families will go elsewhere.
It will take leadership to get this done, but not the kind that won the Cold War. That was hard; this is easy. But we are pussies, and may not be up to it.