Well the problem with your analogy is that illegals are not embezzling. You seem to think their is a finite pool of money for wages in a country. That is not true.
My point is that their illegal presence here represents an ongoing crime, rather than a one-time event. There is no reason why their crimes should be rewarded with more rights than are received by people who file paperwork to come here legally.
Besides, some hospital emergency rooms and other facilities might very rightly regard the illegals as stealing funds from citizens and legal residents.