We have better and cheaper ways now, a simple national ID program required for jobs (with teeth against employers) would have them leaving at near zero cost to us.
Respectfully, this will not work. A large portion will stay anyway: The construction industry, etc., will continue to pay them low wages under the table, en masse; like prostitution, most of it will not be found nor prosecuted. Other undocumented immigrants will stay, leeching off their American family members. Others will pervade illegal "industries" to make a buck. And the remaining will come and go from the country, almost as they please because we stopped taking border security seriously, you know, 'cause we have an ID system now, the one that gets hacked every other year.
Money is better than poverty, if only for financial reasons.