Please check this June itinerary for Japan

A day trip from Kyoto to Himeji Castle can be very rewarding. Watch James Bond in "You Only Live Twice".

You can arrange for a free English tour guide: https://i-guide.jpn.org/

The guide will not accept money. A small gift will be appreciated (we gave a bottle of maple syrup and offered to buy lunch).

I would add a second night in Hakone so that you would have one full day there.

Consider a trip to Nikko north of Tokyo.
Also, Nara from Kyoto

I second the suggestion to do a day trip to Himeji (this was my favorite castle we saw). We did this as our first full day in Japan after spending the night in Kyoto, and it was very relaxing. We had planned to do lunch in town and then head back, but the restaurant we wanted to go to did not have A/C, and found this to be the case for many places in town. We took the train back to Kyoto and found a place that we enjoyed that also had A/C.

Kyoto - If your family is okay getting up early, you will find that you will have many of the "busy" tourist locations to yourself. However you may also miss out on goshuin stamps because you are there before that part of temple/shrines open. It was a good trade off for us.

Tokyo day trip suggestion - Mt. Fuji, we took a bus tour to Lake Kawaguchi area, where they offer a hop on hop off bus for local sights. The Itchiku Kubota Art Museum includes a collection of vintage Kimono's (you aren't allowed to take pictures) that are absolutely beautiful!!! Bonus content - take the bus in reverse order to do the tram to go up to see Mt. Fuji to beat the crowds.

Himeji Castle was great, I strongly recommend it.

There is a lot to see in Osaka and you might consider 3 nights in Kyoto and 3 nights in Osaka. Kobe is another day trip from Kyoto/Osaka.

I also recommend Himeji - one of the few standing original castles. It is massive and you do walk and climb a lot inside.
I visited it on my way back from Hiroshima to Kyoto. I stopped at Himeji (left my daybag at the train station locker) and spent a few hours at the castle and nearby gardens. They had free guided tour as you entered, but I opted to explore solo.

An advice I have for your Japan trip is to bring a few disposable slippers with grippers (or a few you can wash). I got a few at my local dollar store. As you will see, most sacred places require no shoes. Himeji floor was very slippery, and people walking around with their socks struggled.

When in Kyoto and if not a rainy day, I suggest heading north to Kurama/Kibune. I did the hike from Kurama to Kibune, then explored and had a late lunch at the quaint Kibune village. But you can just visit the Kibune village and temples and skip the hike, and dine on slabs they put over the river. Lots of restaurants with traditional japanese meals. The most unique experience was at Hiborun restaurant in Kibune, where you eat nagashi somen (they put cold noodles through a bamboo). It is popular and only open during summer when not raining, and long wait so if that is of interest, go early to get a number/wait, then go explore and return at your designated time.

https://japanjourneys.jp/kyoto/kibune/experiences/nagashi-somen-hirobun/

I second the suggestion to add a second night in Hakone. It takes a while to get there, even though it is not far from Tokyo. It would be nice to have a full day to explore.