Let me take a run at it. A drivers license verifies that the person to whom it was issued is a competent driver. Whether or not he's a citizen, or who he says he is, isn't really germane; the only question is "can he, whoever he is, meet the standards set for being a safe driver?"
So the ID part of the license (except as it shows the person holding the license to be the one to whom it was issued) is extraneous to the license's function. The only reason to have it is if you're adding ID as a separate function... and if you are, then you are going to want to be able to require people to show that ID on demand. Why else have the thing?
Suppose we just got rid of the driver competency aspect of the thing and required every state to issue an ID card that certified the bearer had passed a background check and was who he said he was. Is there any doubt that a demand to see that ID would quickly become the first step in any dealings with the government, including those the government initiated?
It's also a verification of identification - has been for some time.
Not true. In Texas and probably most other states, the address listed on the drivers licence must be the real physical address of the person to whom it is issued. If someone specifies a phoney address, it will not be delivered and it will be sent back to the DPS. The last time I got my license renewed, I also had to have both of my thumbs scanned on a fingerprint reader. At least in Texas a drivers license not only is a document stating that the person is qualified to operate particular types of motor vehicles with particular restrictions but also a document verifying that the person is who he says he is and lives where he says he lives.