IMO that connection is tenuous at best. If you want a connection to violence, look to alcohol.
But it's a slippery slope, particularly if you believe in the right to bear arms.
I admit I haven't gotten as much flak on FR, a pro gun site, as elsewhre, but I've occasionally posted here on the disingenous nature of the ban on, in Illinois and elsewhere, gun ownership for those committed to mental institutions, and everywhere the ban on purchases.
"Crazy people", like felons, sure, there's a legitimate public interest in regulation.
But most affected are alcohol and drug addicts, who seek treatment.
A LEO with an alcohol problem, everything is OK till he gets help, then his job is in jeopardy. For the LEO, his union will take care of it.
What are you going to do, ban gun ownership for people on antidepressants?
The clear constitutional issue aside, if a gunowner has a problem, what do you want him to do, seek treatment and surrender his guns?
Doesn't take much imagination to see where that goes.
There is already a drug question on the Form 4473. Question 11e:
Are you an unlawful user of, or addicted to, marijuana, or any depressant, stimulant, or narcotic drug, or any other controlled substance?
A person taking Prozac, Paxil or similar SSRI is by definition addicted to a controlled substance. Once you start taking an SSRI, it is very difficult to stop. Careful medical supervision is required to prevent triggering a psychotic episode from sudden withdrawal.
It wouldn't take much for a lawyer to argue that a person taking an SSRI who answered NO to this question was committing an act of perjury. Especially if that person subsequently committed a criminal act with the firearm acquired via the Form 4473. There are plenty of other mass shooting cases where the shooters were on an SSRI. Lots of fodder for litigation and legislation.