The flip side is that it's Civil Service, meaning everything is scored and at the end of the day the hireable pool of applicants is typically 2-5 vetted and certified applicants for a given position, so the official hiring body (typically the township supervisors or council) has very little decision-making power. In my state they're legally required to hire one of the top 3. If a veteran is #1, the hiring body must hire them unless they can find a disqualifying feature in the interview. These requirements vary a lot by state, this is just my anecdotal experience.
Even in the smallest local police forces - the smallest I worked with was 8 officers, mostly part-time, in a borough of ~1,000 households - it would be unheard of not to get some sort of psychiatric or psychological opinion on the candidate.
A dude from my high school, whose father was a SWAT lieutenant, tried to join but was denied. They had interviewed a bunch of his friends, and during the interview he was asked "why do you yell at your girlfriend so much?"
A smaller department, lets say Bakersfield PD, might not be so picky.
Will it lead to a small increase in the number of shot police officers? Probably. I would rather have that than the current situation.
So, no, you're wrong.
I don't believe this is true.
I imagine it depends alot on how the upper management enforces discipline and the attitudes they pass down to subordinate officers