While you do some adjusting, I'll add a few things for your consideration.

The Philosopher's Path is really just a long slog that will eat up a lot of your limited time with very little payoff. Some of the little art galleries are nice but I doubt they would interest a youngster. I much prefer to just grab a cab or local bus, hopping off to see Ginkaku-ji, Nazen-ji, the pottery slope (Sannenzaka and Ninenzaka streets) and Kiyomizu-dera. Save Yasaka-jinja for when you visit Gion. It is right across the street and is actually the patron shrine for the Gion residents. It has little for tourists except interesting buildings and grounds but is likely to be quieter than many of the more famous temples.

Frankly, given a choice, I much prefer Kinkaku-ji (the Golden Pavilion) to Ginkaku-ji (the Silver Pavilion - which isn't really silver, or even finished.) The garden path around Ginkaku-ji is nice but the facilities are underwhelming. There is a nice albeit touristy shopping street leading up to Ginkaku-ji that has souvenir shopping, snacks and a few restaurants so YMMV.

If you decide to work in Kinkaku-ji, Ryoan-ji is quite nearby and one of my favorites. The Zen Rock Garden is interesting and evocative. If you are lucky enough to have early cherry blossoms, the walking path around the lake at Ryoan-ji is spectacular.

I don't think you have time in your limited visit but a day trip from Kyoto to ancient Nara is one of the signature Kyoto activities. For an 8-year old, the bowing deer (actually begging deer) in Nara Park, near the giant Buddha at Todai-ji, could be a fun change of pace. I don't know if the deer would be more fun than the monkeys near Arashiyama but you realistically only have time for one.

I agree with others that you have a lot of temples and shrines. The ones you have are nice and quite beautiful but I suggest breaking them up with other sights in between, such as the outdoor Nishiki Market for some interesting walking.