rkold paints a very good picture, I think. I'd forgotten what it's like to travel with kids but mine at that age would have staged a mutiny if I'd made them sit in a car for hours on end every day on what was meant to be holidays.
I'd suggest three hours on a train is a lot more fun and a lot freer (you can get up and walk around, go to the loo, etc) than driving. There's a lot to see out the window, and people watching to get on with. I appreciate the changing looks like a lot of fiddling around but the driving scenarios are best case, which aren't always the way the day goes. Whereas the trains will be there when they say they will.
You might have a better time jumping on the train to Kamakura from Tokyo for a day trip. Then head to Hakone, stay overnight, have the day there then doing a loop around the area and stay another night somewhere else (I haven't been so can make no suggestions about how to do that). After the two nights, go back to Odawara by train and catch the Shinkansen to Kyoto. You could stop in Shizuoka for the night here if you wanted to.
Stay in either Kyoto or Osaka (as others have said, don't bother changing hotels, it's easy and quick to train in between) and do day trips by train to Nara, Himeji, and either Kyoto or Osaka, depending where you aren't staying. Take the Shinkansen to Nagoya. You could have a night there if you like, or head directly from there to Matsmoto for a night or two. You might want a car in Matsumoto to see some more remote things.
I'd then head back to Tokyo, maybe stay a night, or not, and head to Nikko on the train, maybe get an overnight package. You could hire a car to go to the waterfall if you want to.
You then wind up back in Tokyo ready for your flight home.
On another note, you're doing more moving around than I would choose for that length of holiday, but everyone is interested in different things. If you spend a day or two in Kyoto proper you could drop Kamakura (which I love, but if it's your first trip the temples and shrines might get a bit wearing, particularly for the child) and add that day somewhere else. Similarly, while Nikko is lovely, it's also a lot of temples and shrines. You might enjoy fewer destinations more.
If you were interested in considering this you could use this calculator to see if a JR pass pays off.
https://www.japan-guide.com/railpass/Enjoy the trip!