Yes, you are missing something, any illegal alien can apply for a driver lic. in any state and be instantly registered to vote. This law opens the door to vote fraud.
I don't see how this changes things. If an illegal alien gets a driver's license, he can use that as ID to register to vote after illegally and incorrectly verifying that he is a citizen. What makes you think that someone intent on fraud would be deterred by having to take two steps to do it rather than one?