As to the registering it like a used car - it already is. When a private individual sells a car to another private individual, they only sign a quick couple of documents - title, and BOS. There is no background check.
youre legally required to register the car in your county or face penalties and fines/jail time. The car is not allowed to be driven after 7 days or face penalties and fines/tickets. Possible impound. Also, youre mixing the two scenarios. You cannot be denied a NEW CAR because you have a DUI, you can be denied a LICENSE or INSURANCE which prohibits you from driving a car correct? There is no "background check" when you purchase the car new, so there is no Background check when you purchase the car used.

Guns- We require background check to purchase NEW, we should require background check to purchase USED.
You cannot place a requirement of a background check on private sales. The only alternative is to have all individuals get a registration card allowing them to purchase a firearm. Register the person, rahter than the weapon. That would be an infringment on the individual's rights from a legal perspective.
Why not? What makes the USED sale different from the NEW sale? Same firearm, same serial number, same danger/respect/caution needs to be used. Why shouldnt the same requirements be used?

What you suggest is plausible too, and im actually more in favor of that road. Require people to get a FIREARMS LICENSE that is some standard we all agree too. Certain level IQ (above mental retardation) , psychological test, safety course, firing course. No different than what we do for drivers licenses. Retest every 5 years. you could integrate it into the DMVs.

But i figured my route was less intrusive, silly me