The problem is that not having a license does not stop people (of the usual legal age) from driving. Think of all the people who have been stopped for driving on a suspended license.
I would rather know those who are on the roads have been tested for the knowledge of traffic laws and driving ability. I think of this as a public safety issue. Such a license should be temporary of no more than 6 months duration. Maybe that is a way to make sure they are not a threat to other drivers yet gives ICE plenty of time to find them.
I mean the applicants will be reported to ICE, right? Making sure other drivers are safe does not mean ignoring immigration laws completely.
Your reasoning is straight out of the liberal play book. This has already been tried in other states, with bad results.