You got it.
Except for the part about "illegals do a lot of the work most Americans don’t want to do". According to the Law of Supply and Demand (applied to Labor), wages should rise until the market responds with sufficient workers to do those jobs. Hell, even I, a lazy, privileged, self-indulgent and over-protected 1970s hippy-dippy college student was willing to work construction in the hot Florida summer because the pay was decent.
It you let the Free Market operate freely, wages will rise until the supply meets the demand. Isn't that the way the system is supposed to operate?
So if the price of tomatoes goes through the roof because we aren't relying on Mexican slave labor to pick them, then so be it. That's the Invisible Hand at work.