A lot of countries have mental health response teams that work with the police to balance the safety of the individual in crisis with public safety. Seems like it’s worth looking into.
Many if not most large cities already have this or something like it.
Also, I think your implicit assumption is incorrect (that the police typically don’t have the individual‘s safety in mind, or that they only have the general public’s safety in mind).
My implicit assumption is that rank and file police officers regularly lack the years of psychological training required to appropriately respond to these kinds of incidents.