That assumes that these workers are coming from countries where opportunities for paid work are equal to or better than those available in the US, and are equally lucrative. Many workers come from communities destroyed by the drug wars, or from economies ruined by corrupt governance. If those structural factors don't see improvement, the impact on net low-skilled migration to the US won't budge much.
For a lot of people, decision making is based on which option allows them to just live another day; relative to their circumstances, we are exceptionally privileged to be able to make decisions based on the extent to which macroeconomic indicators impact our standard of living.
And, as minimum wages go up, high school graduates become more willing to work. At $10 you may not find manual labor worthwhile, but at $15 it looks more enticing. Now, immigrants typically work a little harder, but not so much to overcome a lack of education and poor communication skills --which can take years to develop if Eng is not their native lang.
As tech advances and "hollows out" low skilled labor from advanced economies, these economies will struggle to keep their mid skills labor market employed.
Too few skills (reading, maths) and now automation is coming to agriculture even fast food restaurants, and at $15 min wage, you can hire a high school dropout, if you really need manual labor.