I believe you have a good point. There are several fingers to point at the cause of the problem. You listed another, and imho, valid one.
I see it occasionally at one place I work. Those who come in for pre-employment screening, with obvious (to an amateur eye) fake or forged documentation, roll in from time to time.
Is there any way I could anonymously *report* something, if I suspect anything? Is such an idea realistic, with the magnitude of the problem?
I would start with a local dept. of labor, they may have some kind of tip line.
Most local/state governments have some sort of employment fraud prevention policy, although I'm sure results vary.