That's strange, it seems like the chain and gear are fine, but something at the shaft is slipping and catching.
How about one more video? Remove the bolt that holds on the gear and get real close to the gear.
I was going to post yesterday, but I thought this was too much of a long shot.
When I took apart my transmission I noticed a few things. I made a few videos of it. The drum holds tension on the fork and the fork pushes the gears together.
I'm thinking you either have a sprocket/shaft problem or a transmission problem where the gears aren't going into place.
Here's a video of the play in the clutch. That's the clutch basket and you can see the amount of play in it.
When you shift, the gears have ears have to line up with the other gear. They wear over time and are harder to get them to mesh.
Q. once you are in a given gear, is everything 100% fine? Meaning, if you go into 2nd gear and you accelerate / decelerate several times, does it make any noise or anything?
If yes, then I'm guessing the transmission isn't meshing. It would be telling to see the shaft where the sprocket goes.
One other thing you can do. Get some black and white paint. Paint the shaft tip and sprocket black and mark a line on both in white so that we can see if they are moving in relation to each other.
You can also use some sand paper and some red or yellow paint and make a thin line or make an "X" so that when it does it's thing, we can see the movement in relation to each other.
Don't forget to have the camera as high res as you can.
I don't know the history of your bike, how long has this been going on, what's the history of the sprocket and chain? Is there any chance the sprocket and chain don't match or there's some odd damage to the chain such that it is loose at the spinners (the inner drum that contacts the teeth). Can you move the spinners, do they have excess play?