Why Do All Police Routinely Carry Guns?

Police shootings of unarmed persons have been all over the news lately, most recently a mentally ill homeless person on skid row in Los Angeles.  

In almost every case what seems to precipitate the shooting is the officer perceiving that the suspect is trying to grab his gun.

It has made me start wondering why every police officer carries a gun, particularly in urban settings.  In the case involving the LA skid row killing it appears there were 5 to 7 officers on the scene - what if some of the officers were armed only with non-lethal weapons and were charged with actually engaging suspects, while officers with weapons monitored from a distance where no one could grab their gun?

I would have officers having only non-lethal weapons be an elite force, chosen for their social skills, athleticism and fitness and they would have special training and pay.  After all, guns aren't heavy and the most out of shape person in the world can shoot someone.

Police in England have generally don't carried guns except in special circumstances, and seem to have done just fine.