Oh!! They are available, it's just easier to sit at home and collect the check than get up and go to work.
You are exactly right! I cashed checks at the local bank from Mexican workers and seems like they earned a very decent wage. Also they lived rent free in the packers quarters. They did have to pay for food, but most of the money went right back to Mexico. Transportation was free from Mexico but they needed to pay the return fare.
I was amazed at the number of women willing to leave behind a family for 9 months to work in the US. After 2 or 3 years of saving the earnings, they could live quite well in Mexico for a very long time.
So CIS admits that unemployed workers presently don't have incentives to take jobs. Any adult male who doesn't already have an incentive to get a job won't benefit from kicking out foreign laborers.