I’m hearing a lot of shouting in that video, but no “I’m going to the police”

Try turning your sound on. It's clear as day. Gaige runs up to Kyle and says "You just shot somebody?" and at 0:11 Kyle responds "I'm going to police". This is not in any way one of those videos where it's unclear or open to interpretation, and the only way you can watch the video with sound on and claim not to hear that is if you are lying.

If the protester didn’t have his gun drawn, and Kyle did, then Kyle becomes the provocateur again.

Nope, because once again, Kyle lowered his weapon as soon as the threat ended. At which point Gaige once again attacked Kyle. Again it's clear as day on video, the second Gaige puts his hands up and backs away Kyle lowers his gun.