"A universal background check would have directly affected what
happened here in Tucson," Mr. Kelly said, referring to the shooting in
which six people were killed and many others, including his wife, were
injured. The gunman, Jared L. Loughner, had been suspended from
community college for behavioral reasons.
Being "suspended from community college for behavioral reasons" is not the same as being "adjudicated mentally defective". While Loughner definitely qualifies for that judgment in hindsight, he was not on the prohibited person list when he bought his gun. Furthermore, he bought his gun from a licensed dealer who did in fact run a background check.
Making background checks "universal" by banning private firearms transfers would not change anything, because Loughner was able to pass a check. If the people who encountered him and felt threatened by him had reported it, perhaps he would have been on the prohibited list -- but perhaps not, since "being creepy" isn't a felony and a mental health disqualification is a vague standard.
Spree killings by the mentally ill are rare and horrific events. They are an argument for attempting to improve our mental health system, so they can be identified and treated before they kill people. But identifying the mentally ill who will become violent in the future is not an easy problem, and relies mainly on identifying those who have been violent in smaller ways in the past. Some people will always slip through the cracks, because in a free society we cannot lock up people for what they might do in the future.