How old is the bike, or better yet, how many total miles do you think is on it. Though everything else in your pics looks nice, clean and almost new, the pulley wheels in your video looked a little worn... maybe, to hard to really tell. So if they are worn, then that might be the reason for the skip.

Or if that is the old chain on the bike, then it may be worn and stretched (so to speak) and that might be why it is skipping on the pulley wheel.

It's pretty easy to measure a chain to see if it's worn. Put a metal scale next to it and see if the 12 inch mark falls on the same part of the pin as the 0 mark is. If that part of the pin is a 1/16th beyond, then the chain is at or near replacement time depending on your preferences for replacing chains and whether you want to limit how often the cassette and rings need to be replaced.