because he had to show everyone that he was in charge

That's exactly it. Darnella Frazier challenged him, simply by being there with a camera. The implication of the camera was "what you're doing is wrong, and I intend to expose you". Chauvin responded to that challenge. His actions essentially said "I decide what's right and what's wrong. I'm going to kill this man in front of you, to show you that I can. I'm going to get away with it. If you try to stop me, I'll kill you too."

That's why he stayed on Floyd's neck for two whole minutes after Floyd went limp. It was about showing everyone that he is in charge, that people die when they don't do what he wants. George Floyd was Chauvin's 4th murder, he had every reason to believe he'd get away with this one too and before we all saw the video he went straight to the playbook that helped him get away with the other three. Without Frazier's video, Floyd would have been exactly what Chauvin said he was: a counterfeiter who died of an overdose while in custody. Before Floyd, Chauvin had pistol whipped a domestic violence suspect and then claimed that suspect reached for his gun, had shot another suspect after witnesses said that suspect had dropped his gun and put his hands in the air, and beat an unarmed 14 year old boy across the back of the head with a mag lite so badly that boy required stitches. He liked to make money off duty acting as a bouncer and the club owners that hired him said he did things like pepper spray the entire crowd in response to a fight breaking out.

Derek Chauvin is a monster who only knows violence. If he ever gets out of prison, he will kill again. Killing is what he does.