It is black and white in this case. Pay a fair wage or don't hire, and don't hire people that are in this country illegally. How hard is that? If (general) you can't afford to pay what Americans consider a fair wage then your business isn't going anywhere.
People pass up jobs when the wages don't reflect the effort put into doing the job. Not because the people think they are too good to settle for that job.
You complain about blanket statements being made, but you're the one making them. You seem to think Americans are snubbing less than proud jobs that we think are beneath us and that's just so not the case. Illegal immigrants being hard working and loyal because they're so happy to be here doesn't really fit into this debate. Nobody ever said they didn't work hard.
Being willing to work for less than the value of the job skews the market, no matter how nice and loyal and hard working someone is or how good their intentions are.