A significant portion of the US food supply is produced by workers here illegally. A Google search for the phrase "ice raids" will produce numerous accounts of large food processing operations having to shut down for weeks because a large number of their workers were caught using fake SSN's. If this labor went away food prices would have to rise significantly to cover the higher pay that would be required to attract workers to the agriculture industry.
One puzzle is how the suppliers of fake identification determined that there was a market for social security numbers among migrant farmworkers. There had to be some communication with employers because it wasn't that long ago that I-9 files (employer proof that they asked for proof of legal work status) weren't being checked regularly by federal authorities. Now, when a business gets raided by ICE the employer acts shocked and complains to the press that they have no workers.
I think that agriculture is a good example of the classic labor/capital value issue. Without workers, most farmland is virtually worthless. Some crops can be produced with heavily mechanized systems, but most require workers to supply the freshness and quality that US consumers expect.
Historically fresh vegetables and fruit were something you grew yourself or went without for the average person.
It would easily be possible to source fruits at vegetables overseas and eliminate the pull for illegal farm labor; but that would cause economic distress for farmland owners, who are politically powerful. California is a state where agricultural interests have a strong influence.
In one of the early waves of globalization, English landowners suffered significant economic losses when cheap food could be imported profitably to England due to improved transportation systems that allowed land that was cheaper overseas to be used for agricultural purposes.
It might be possible to block labor flows, but that would invite increased imports of food products. Closing all borders would invite smuggling. Having a balance of some immigrants and some food imports is probably realistic.
The same capital-owners are driving both of the negative forces that are causing the enduring high employment and leading to other negative economic outcomes. If there were
a little less capital income, and a little more labor income that would result in a better distribution of resources to restore economic circulation.
One thing the US could do is drop subsidies for foreign food aid. This would increase both the cost of food and labor globally while allowing virtuous cycles based on local agriculture to develop in foreign countries.
Also, consumers could save themselves a lot by purchasing food in bulk and doing the processing at home; i.e. cooking. Heavy processing and marketing is where a lot of the price of grocery store food comes from.
US agricultural policy should favor crops that can be sown, grown, and harvested by mechanized processes. This would result in higher-skill jobs maintaining the equipment and operating it.