They can drive down wages to the minimum wage.
But most people are missing what an employer has to shell out to employ even a minimum-wage employee:
1. The other half of the FICA/Medicare payment.
2. Overtime when the employees are on the job longer than 40 hours per week.
Employers who have been paying illegals under the table, or, at best 1099-ing them, are going to see a substantial increase in both paperwork and employment costs when these illegals become “legal” employees and start demanding legal wages.
And I’m sure the various grievance groups will create programs to report employers who aren’t paying the full wage, and unions will rush in to unionize many of these employees.
So now employers will have a choice: employ someone who can’t speak English and follow directions easily, or someone who can. The costs will be the same. That answer is pretty obvious, especially on construction sites.
The problem with your theory is that it assumes there are plenty of Americans willing to work those jobs at minimum wage once the illegals drive the prices down to that level.
If liberalism has proven anything it has proven there are plenty of native born Americans that would rather sit on their behinds and collect a government check rather than work minimum wage. I don't see how driving unskilled wages down to minimum legal helps anyone but the employers, and that benefit is only short term up to the point where many unemployed Americans can't afford their services anymore.
I'm back to my original point. Illegals will have to be driven out of this country by force. They won't leave up to the very point where they have to and they have a distinct advantage in the workplace because they are willing to work for much less money and benefits. They hold that advantage regardless they are here legally or not - and you can look around everywhere and see this.