Would you find an updated version of the Bracero program acceptable? I.e., issuing permits to WORKERS to come to the US, and then to leave. They do NOT bring their families. They come, work the agricultural seasons and return home until the following year. That program seems to have worked fairly well until it ended in 1964 (the beginning of our “Great Society”)
IMO, much of the problem is that the US has offered too much incentive for folks to come here, en famille. We offer, through Congressional mandates, free health services, free education, and often free food and housing. What’s not to like?
If we removed those incentives, and offered permits to workers from Mexico (or elsewhere) to come for, i.e., a given growing season, with their return home at the end of the season, it would work to everyone’s advantage. (I would include some medical coverage provided by employers for on the job injuries).
With the proper controls (to guarantee illegals do not take jobs from American citizens), I would have no problem at all with another Bracero program. No anchor babies, no tapping US taxpayers, go home when done.