I agree with you that this is primarily an issue of wage rates, not demeaning work. Just as a matter of curiosity, I would like someone from one of the states that does not have a large population of illegals to comment on who does the low level jobs there.
Here in Texas, I see them in all types of jobs, including construction, factories and others that were once relatively high paying positions. Possibly these type jobs don't exist in all parts of the country or are just not being filled.
Drive down Chapman Blvd in Orange, CA, around noon, and you'll see crowds of obvious illegals lounging around, some waiting for a pickup truck to collect them, others just lounging around. What you don't see are fields of crops needing picking, or orchards needing pruning. The illegals are doing the work that Americans will and do handle, if paid like they once were. Hats off to the illegals who work in agriculture, but they are only a small percentage--and their multiplying kids, wives, girlfriends, grannies are definitely not out there in the fields and orchards.