Good theory, but you need both.
No regime change: right on, but it's the corporatocracy that pushes for both that and illegals' labor. NAFTA: actually reduces the pressure. No subsidy flooding: right on.
Employer sanctions: been in place for years, just not enforced that seriously when they can just de-fund the enforcement.
The REAL fix would be undermining (1) the pressures created by Rome's war on birth control and (2) the corruption and graft in those third-world countries that make it hard for them to live at home. The best measure of true compassion for the illegals would be those two right there.