I don't think it's capture.
In a free market, businesses are supposed to use regulations to their advantage. If the regulations don't achieve their goals, legislators are supposed to change them. The latter part often doesn't happen in the US. Not because the Congress is captured by business interests, but because it's dysfunctional.
And the society as a whole is somehow unwilling to admit that the US needs large-scale immigration to function. Just like almost every other developed country. At the high end, every developed country is a net importer of talent in industries where it's above the average (relative to the size of its economy). And at the low end, there are essential jobs that are inherently too bad for most citizens and too demanding for most of those who can't find better jobs.
The best practice these days is making temporary work permits industry-specific. If there is labor shortage in an industry, just let in anyone with the skills and no obvious red flags. And if there is labor surplus, only let actual talent in. If an immigrant loses their job, they can try to find another in the same industry without excess bureaucracy. And then give a straightforward path to permanent residency for those who have been in the country for N years without causing too much trouble.
Illegal immigrants largely work in fields such as agriculture, construction, and maintenance. The pay is inherently low, because the fields are labor-intensive the outputs are expected to be affordable to most citizens. Many jobs are seasonal, and many require you to live near the worksite (instead of with your family). Most are physically demanding and require a good work ethic. Once a country gets rich enough, it usually discovers that citizens who can't find better jobs often don't have what it takes to do these jobs.