I just did my first kart track about a month ago and it was fantastic, it looks very similar to your videos. What a great way to practice and work on those skills! Hitting the apex, working on lines, experimenting with lean angles, and braking, it's all good. It looks like you guys are having a great time.
I rode a drz400 with slicks but it was such a good experience it inspired me to set up my Dad's first gen Ninjette for the kart track. I'm probably 15 minutes from a suitable track, or a half-hour from the one I tried last month.
Keep having fun, it sure looks like a blast