Meanwhile the illegals can't complain about working conditions - and will work for less than minimum wage in (those occupations where it applies.)
Well, pickers are often paid by the amount they pick, rather than simply an hourly wage. The reason your average young American can't make decent money is because these are SKILLED LABOR positions. It often takes a few years of picking a particular item of produce before you get enough experience to do it most efficiently. Many pickers specialize in certain fruits or vegetables; hence why many of them are "migrant," since they follow the harvest of what they're good at.
The problem isn't that one can't earn more than minimum wage doing picking -- it's that most Americans view picking as a temporary job or summer thing that they'll do until they find something better. But you have to do it for quite some time before it becomes profitable.
You might read up on what happened in some southern states that passed laws to make it more difficult to hire illegals. They still had migrant legal workers who were pros and could make money, but most of the Americans they'd try to train would quit in a week... It's hard work, and unskilled workers can't keep up enough to make decent money.