I have no idea how the Work Visa program works.

I have no idea how many Work Visas are issued every year.

If the program is too cumbersome and/or time consuming, then revise the program.

If all foreign workers were vetted and registered through the Work Visa program, who would have an issue with it?

But just saying "We need someone to pick our cotton" so that justifies illegal immigration?

Nope.