There is room for "government involvement in a free market" and it's unrealistic with any civilization to suggest otherwise. It's the DEGREE of "government involvement" that most of us object to but it's unrealistic, even reckless, to suggest that there be no government involvement.
If and when the "government" is acting on behalf of what's good and beneficial to free markets, that is.
The only "free market" that works "fine" without government involvement is crime. They really enjoy their freedom to do whatever without government involvement.
I'm not suggesting a top heavy "government involvement" to solve the illegal immigration problem and suggest you don't try to replace my good intentions here with corrupt, big government control type violations of what we generally mean by free enterprise.
Without some government involvement, however, the problem won't be solved, as with many others that affect huge areas of our population and economics. There are always cheaters among the human population and I regard "free enterprise" that encourages breaking laws to be cheating. Most Americans support existing immigration and citizenship requirements and the chief "cheaters" involved here are illegal aliens and the people who use them in the U.S. I can understand that to them,, any government involvement is too much but so it is to most who are involved in crime.
I know individuals can't organize their own law enforcement and military. Nor penal systems. There are areas of involvement in free enterprise that require government acting on our corporate self as a civilization and citizenship and immigration are some of those.
You make some good points.