The point isn't to prevent a skilled attacker. The point is to prevent casual lying and low-skill fraud. Most people who lie/cheat/steal do so because it's easy or because they're dumb. Your QR code idea will cost more money to implement and won't block skilled attackers either, as it doesn't take a genius to figure out a way to get mail from a mailbox you don't own.

Utility bills are the DMV's equivalent of a cheap lock. A smart attacker can pick the lock, and a determined attacker can cut it off. But the majority of thieves are walking around looking for unlocked car doors instead.

Do you really think DMV asks for a copy of a utility bill because it's a good way to suppress your rights? I would think that there are plenty of more effective ways to do so, if that were actually their goal.