Thanks to Roe v. Wade, people have a right to medical privacy, which extends to psychiatric treatment. Unless a therapist believes their patient is an imminent threat, they are forbidden to contact the police. A history of mental illness rarely shows up on the FBI background check. The only information is provided by the purchaser in a check box. If they say they are not a drunk, druggie, or lunatic, their approval sails right on through.
Not that the United States provides treatment for mental illness. It's an invisible affliction. We have all met our share of crazies. Most of them are harmless. Some are not.