Government records are a matter of life. Without them, it’s not possible to keep track of simple things like who is authorized to drive, who’s the legal owner of a given piece of property etc.
So long as there is a simple mechanism to correct the database being used (for example, show your US passport or birth certificate to prove you’re eligible to work), we should move forward with a sensible bill like this.
Driver’s records and property ownership records have never been the province of the national government. Those functions have been exercised at the state and county level.
Can you point me to any portion of the U.S. Constitution which empowers Congress to pass laws dictating who may or may not work?