You constructed a narrative and overlaid it over the facts.

The facts are that for many decades past, it was possible for hopeful economic immigrants to abuse asylum laws, or the back then less protected border, to gain entry to the US. Neither red nor blue administrations handled this properly and lots of people benefited from the status quo, and your focus on, "the left" is quite conspicuous, because one does not tend to think of farm owners, meat processing plants and construction contracting businesses as "the left".

And rightfully past administrations should shoulder the blame for not dealing with immigration in a lawful manner back then. If there was a need for immigrant labor they should have handed temporary VISAs or whatever, instead of ignoring illegal immigration.