It would be nice if a background check included psychological illness, which I don't think it does. Someone correct me if I'm wrong, it only checks if you're a felon or have any outstanding warrants? You would have to sign a waiver to waive medical record privacy. Surprised this isn't already being done. Like others have said a lot more needs to be done outside of gun control but it would be a step in the right direction.
Trying to include a psych eval would cost a lot of money and as someone who has taken a few of them for employment as a sworn firefighter I believe they're very subjective. Better to just access medical records.