I see what you're saying, but I still think the market itself should block the entry of people not in demand. Even if they are not in demand where they are, why would they pick up and move someplace else where they are not in demand?
My opinion - fear in the US is not for the lettuce pickers taking jobs from Americans; it's about the jobs in construction, manufacturing, etc. where skills are in demand.
So - there's a fear that people with no skills will come and abuse the social safety net - but to me that is a reason to fix the safety net.
Second real issue - these things don't work as well where they are one-sided. I understand that. People come here, but then if their skills are no longer needed, they can't leave so easily because other countries' policies are tighter... then we are stuck with people with skills we don't want. I'm not sure how you fix this.
But for the market to block entry US society would have to allow what happens when a product is not in demand. I don't think anyone is going to let people starve in the streets, nor should they, but that's the logical mechanism if we're talking free markets here such that they will provide the incentive not to enter.