I've served on multiple commissions whose sole purpose was to interview police applicants (and serve on one currently). In every instance there was an in depth background check including interviews (former significant others, neighbors, teachers, etc), financial background check and credit reports, and a psychological evaluation. At many instances the candidate could be removed from the process.
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.