I understand basic economics so of course see your argument; I'm just telling you what I saw on here. IMO it depends on how far the limits are set by the government. If minimum wage was set at say:£50 ($90)plus social security etc, then I have little doubt that the underground activity that you describe would occur. Of course, it wouldn't get to that as such a rate would cause a national outcry and crisis if the government persisted.
IMO the current rate should not be forcing employers to employ illegal immigrants; as far as I'm concerned if I couldn't obtain an acceptable rate of return if I paid the current minimum wage, I shouldn't be in business!
Rest assured, the fact that you don't see it doesn't mean it isn't happening. If it's a "natural minimum" like you seem to describe, then the government can remove the legislation, and the wage wont drop. Anything else is causing higher unemployment and increased illegal activity.