AZ does not "[permit] anyone over 21 to carry a concealed weapon without a background check." The law still prohibits crazies from possessing firearms. The change was to end the requirement for people who could already otherwise lawfully possess weapons to obtain a permit to conceal that lawfully possessed firearm.
Based on the facts at hand now, what the Post proposes would not have altered what happened one iota. It would just make life more difficult for those of use who don't intend to ever shoot a congressman. No upside. Downside is further restricting Liberty, which, no doubt, is what the Post really wants.