Many, maybe even most illegal immigrants in this country come in legally and then overstay their visa, becoming illegal. If the Indian immigrant superstar engineer is here on an expired employee visa, his/her future children born in the U.S. would not be citizens no more than the Mexican farmworker who snuck across the border's children would be.
Estimates are that 10 million are here illegally. I don't think your assumption is correct
The study estimates that the visa overstay population in the United States is at least 3.6 million people, out of an estimated 9 million to 10 million illegal immigrants. (Source)