Back to the High Programmer's Blog
Frankly, the problem is that driver's licenses are far more than even dual use. If they acted just like a library card, basically and totally useless unless being used for the intended purpose(you may check out a book, that's it), then sure, let illegal aliens take the test so they can drive.
However, it isn't like that at all. Your driver's license goes far beyond specifying your privledge to drive. It is used for all sorts of other identification purposes and some of these relate to getting treatment at hospitals, identification for school/college admission, and anything that usually has to do with you interacting with "The System" and especially law enforcement.
So, now the argument is "now we can keep track of them" to which my response is "and they don't pay taxes".
So the next time an illegal alien is in the CT machine for 2 hours and who will simply walk away without having to pay (and HOW will they collect?). You can think about that train spike in your spine and hope you can still walk again after the delay.
Your primary example (hospital care) is bogus. Hospitals don't just take your driver's license and give you care. For emergency care they'll treat anyone, even someone without any id. Withholding a driver's license from an illegal alien won't do anything for that situation. For non-emergency care they're going to want insurance information. Changes to driver's license issuing won't change anything.
Ultimately the problem is that the illegal aliens aren't going to go, "Well, I can't get a driver's license. I guess I'll leave." License or not, they're going to be here. We might as well try to minimize the harm from having the illegal aliens here. Having them get licenses does mean that we at least have a photo on file. We have an opportunity to do a quick background check ("Does this photo match anyone in the FBI's most wanted list"). We encourage people who are likely going to drive anyway to become familiar enough with the laws to pass the test (making people who would be on teh road anyway safer). We even make a small amount of money (since driver's licenses aren't free).