a popular social target, and often have deep pockets and large insurance polices for lawsuits, I suspect they are actually BETTER at filtering out illegal workers than small companies.

Simply due to economy of scale. A small landscaper is FAR more likely to engage in a cash and carry relationship with an employee (legal or not) because he doesn't HAVE the large scale administrative structures and CAN pay an employee in cash. A large corporation kind of HAS to create automated payroll systems due to how many employees they have and as such really CAN'T do cash payroll.