The question of how far cops are prepared to take any risks with their own safety in preference to killing the suspected threat, had me wondering about cases with British cops. In particular one I recall where one disarmed a mentally-ill person who was attacking someone with a samurai sword. This was achieved by 'stepping inside the swing radius of the sword' and thumping the guy.
This happened at about the same time as a widely-reported incident where US cops shot dead a mentally ill woman who was waving a small knife around.
Trying to google the event turned up a surprisingly large number of different cases here in the UK where cops disarmed mentally ill people waving swords about, most of them described as 'samurai' variety, from all over the country (I never knew we had such a major samurai problem), but also, honesty compels me to mention, a UK case where the sword wielder was shot dead by firearms officers, after failing to be incapacitated by CS gas spray.
There must be a way of quantifying such incidents to determine just to what extent US police are more inclined to employ lethal force in order to avoid any risk, and to what extent this actually reduces police deaths and injuries (and at what cost to the public).
The point about the US public having guns doesn't, to my mind, explain the quicker resort to lethal methods when dealing with mentally ill people visibly weilding knives - unless its common for people to threaten people with swords while secretly carrying a gun.