You two are going in circles.
-Yes, background checks prevent criminals and the coo-coo-for-cocoa-puffs crowd from obtaining gun legally.
-No, they don't prevent them from obtaining them illegally.
The question you should be asking is whether or not background checks deter enough criminal activity with guns to validate their necessity. Maybe I'm wrong, but I think it's probably pretty difficult to measure the amount of applicants that were denied a gun via background check, and decided to quit looking for one, versus the people that managed to get one elsewhere.
Then again, I haven't been paying too much attention your conversation, so maybe you already covered that.