I would like to find a private tour for 4; 2 adult and 2 kids

Hey Sochi, this is the best place you'll possibly get this tours: TripleLights
https://triplelights.com/japan/kyoto

Just click on a guide that you like (they have profile videos) and tell them you want this tour. They'll arrange everything for you.

Or you can buy a tour directly, but messaging the guide first is the best in my opinion.

The support of the website Triple lights tours is incredible as well, you can contact them or call them and they are super helpful.

Hope this helps you as it helped me, cheers!