One of the interesting things I learned about becoming a police officer is that in many, if not most, jurisdictions there is not a requirement that recruits be given a psychiatric evaluation when applying. In Georgia, for instance, the Georgia Peace Officer Standards and Training Council simply requires an applicant to "Undergo a physical examination by a licensed physician to determine any physical, emotional or mental conditions that may adversely affect the ability to exercise the powers or duties of a peace officer." I have heard similar things from police officers. In some cases, the psych eval is simply the doctor giving you a physical asking "Are you depressed or crazy?" If you answer no, you're in. That being said, most police officers do not want crazies on the force with them, if for no other reason than for their own safety. As a result, Training Officers are on the lookout for what some call 'woo-woos', guys who want to see the lights flash and the sirens go woo-woo.
Are things changing in this regard? Yes, slowly. Psychiatric evaluations cost money, money that police departments often don't have. Politicians can yap all they want about upgrading standards and increasing training, but that costs money, which the same politicians do not want to pay. Often it takes a YouTube disaster to get them to actually do something.