https://www.treatmentadvocacycenter.org/storage/documents/overlooked-in-the-undercounted.pdf page 11 under "co-responder teams for psychiatric emergency response" but that entire document is pretty enlightening to the problems resulting in these unarmed shootings.
City leaders across the country are considering how to rethink police departments. They have many examples to look to—from brand new pilots to programs decades in the making.'We've placed more and more issues on the plates of officers who are not trained to be a social worker, or to be an...
msmagazine.com
That's a pretty good summation article of what I'm talking about. They also reference other things such as the low amount of actual violence that police respond to, about 1% of 911 calls in major cities.
How one Kentucky agency refocused its officers, saved money — and earned new respect from the community.
medium.com
Another article, this one an opinion piece by a cop advocating for social workers to be included in policing, so it's not even really very controversial for police precincts, more just controversial for their officers who want overtime hours which make up a large portion of their pay and contribute to stress on the job. Also of note when they are stressed the stress-intervention methods built into most departments to avoid rogue officers isn't utilized because of the stigma of needing help to deal with your emotions within a police force.
They specifically mention violence as a reason for police, however like I said, that's only 1% of calls. There have been a lot of cities who have already moved to great social worker involvement that we can gleam data from in order to make the policies intelligently, but unfortunately there's a huge pushback because the "defund the police" puts a lot of people in hardcore defense mode. In the situation that you mentioned with your father that would be a justification for officers being present as well just as backup, but the primary thrust of the initial contact should be one of attempting to help, not intimidation. People having breakdowns are pretty sensitive to intimidation, so it may not take much. This is why social workers are better trained for this and don't have weapons.
I've heard the opinions of a few social workers who do these types of calls and the consensus is that sometimes they do get beat up, occassionally they'll have their hair pulled, they might get punched or slapped or screamed at, but they both survived and the patient was safely transported to a hospital. This almost always happens because people stop taking their medication and then get stressed. With how much that tends to cost these days it's a sad reality that many people who mean well will end up with a gun pointed at them as they're in a manic state and be expected to be the deescalating party of an interaction. Their odds aren't great with a cop but they'll almost certainly be okay with a social worker.