I think that’s a solid plan for track days and turning laps. Low maintenance for all three karts means you and the kids can have more time driving and less meddling. One caveat is that there may not be a 206 kid kart racing class.
The 206 package can be used from single digit ages all the way up to triple digits if one lives that long. For the younger classes, there is a different carb slide and coil to reduce the speed.
Chassis brand wise, it comes down to what’s common/supported at the tracks you run at. I think you said you had a TBKart? They are a well known brand with support around the Cali area. They make cadet and kid chassis too.
So far fb, cl and forums. The only kart shop within 100 miles is not open for in person due to covid. I called sodi [since they are local]a few times buy no answer.
Karting equipment is typically sold by teams that service karters at the track. These tent programs are frequently dealers for a specific few brands that their drivers use. Round here, it’s compkart.
Ask at the series who sells karts. The tent guys can get you new or used, probably.
Until you know what you are looking for and have experience, you could use some guidance. It’s too easy to buy the wrong thing. A team should be able to guide you properly.