1) that every economist on the planet is wrong when they say that an influx of low-wage labor will drive down wages, and
2) that that decrease in wages has absolutely no impact on unemployment?
Think this through: Imagine that strawberries are harvested exclusively by illegal immigrants for $5/hour. Now imagine that all those illegal immigrants leave the country. Are Americans going to be rushing to pick strawberries for $5/hr? Probably not, but that's where your reasoning stops. You ignore the second half of the equation.
Since there will obviously be a continued demand for strawberries, they have to get picked somehow. If there are no illegal immigrants to pick them for $5/hr, then the people who own the plantations will have to increase the salaries to a level that will attract enough workers. Assuming that's something like $10/hr, the end result will be that a number of American workers that would have previously been unemployed will now be earning $10/hr.
As a side note, the politics of illegal immigration crack me up. I'll never understand how the same people who campaign on behalf of low-income people and argue for increased employment opportunity will then turn around and get furious at a proposal that would reduce unemployment and increase wages among low-income Americans.