Luckily I don't live in the US where there's a high risk of being shot by police if you call for help for mental problems or getting put away because of good insurance, but if I did I would pick the police over the risk of getting medicated by overworked and poorly educated hospital-workers. The police might pass you on but that is still a risk I'd willingly take if I had to pick Police or direct to the social-system in the US. Of course I would rather take a loved one to someone I knew would give them proper care in the first place but most that call the police does not have that option or we wouldn't have this problem in the first place.
In other words, the problem isn't as black and white as you appear to make it. The solution is educated police working with a better social system, not sending social-workers to do a job they aren't educated or prepared for. If you look at much better (but still not perfect) systems you will mostly see that even when mentally sick people that are already in the hospital get violent they still call the police as it isn't their job to pacify dangerous individuals. That is the job of a well educated, not over-worked, police-force.
https://www.psychologytoday.com/us/blog/out-the-darkness/201...