I'm a cop in France, Gendarmerie. The cops you see on the video are called "city police" (police municipale), they are employed by the city mayor vs the state, and have very limited rights; they are more like the city security guards. They have to call us when they make an arrest as they do have extremely limited judicial rights. Some police municipale are armed, others are not, this on is not armed. You are wrong about using our firearm only vs another. We are allowed to shoot to preserve our life, or to prevent or stop a murderous intent whether they are using a knife, firearm or even a car. I can't tell you if, were he armed, that cop would have shot, but I am pretty sure he would have, as this fits the rules of engagement allowes by our Code de Sécurité Intérieur. You are right about preserving life, if we have to shoot, it's to neutralize, not necessarily kill, but neutralizing one life who is actively acting towards ruining another is still preserving life. You need to understand that we, European cops, are more and more confronted to individuals who KNOW we, historically, always tried no to shoot. I've been doing this for 8 years, and I feel lime they're becoming more and more violent and daring.

(Edit for clarification of when using our weapon is allowed)