I have a driver's license, yup, but nobody ever checks it. I even have "Please ask for ID" written on my credit cards and STILL nobody checks it.
Besides, there are plenty of illegal licenses available for the illegals. Unless we go to a national ID and make it difficult to forge, a DL is useless as ID (IMHO...).
SOME illegal aliens have false documents, but you are exaggerating a bit. MOST do not.