The best shrine to visit with kiddies, for us, is Heian-Jingu in Kyoto as it has a playground in Okazaki Park just near the entrance. It has a slide, some small play equipment, a sand pit and also an old tram for children to climb in and out.
Heian Jingu opens at 6am (playgrounds are always open.)
HEIAN JINGU SHRINE
京都左京区岡崎西天王町97
97 OKAZAKI NISHITENNOCHO, SAKYO-KU, KYOTO
HEIAN JINGU SHRINE WEBSITE
*Also, if someone in your group has mobility issues or you are stuck for time to see the temples in Kyoto, it is possible to rent an English speaking taxi driver/tour guide. I can recommend booking with sightseeing taxi Hiroshi Taxi (prices listed on this website.)
If you are in need of an indoor option for kids near Heian Jingu Shrine, I recommend popping into the Kyoto Handicraft Centre (6 mins walk from Heian Jingu Shrine). Here there are handicrafts to purchase, restaurants and also a small play area with lots of Japanese toys for children to play with .