> the cop, by definition, must identify and respond faster than the suspect pulling out his chambered weapon

You are correct about what the cop must do here, assuming the suspect pulled a weapon. If that didn't happen, then the cop doesn't necessarily need to do that.

> That doesn't work well for the cop

That is entirely possible (though rare) for a cop. Part of the trade-off of them getting virtually unlimited power to protect others is that potentially saving other people's lives outweighs potentially saving their own lives.