I think the point is that once the DL becomes a uniform document, it will become easier to check for validity. In most cases when such an "internal passport" is created, it isn't long before it starts being treated that way, and being required to move from place to place (either through blatant border crossings, or through "just checking to see if you're legit, by the way, where are you going and why?"). Many in the US are concerned that if such a uniform document is implemented here, it won't be long before the government decides to start using it to monitor folks.
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?