Oh, don't forget the beautiful garden at Heian Jingu!

I'm no expert, but now you say "We plan to do some parts of the Higashiyama walk to visit a few temples, Sanenzaka and Ninenzaka on our own," you can do Kiyomizu yourself, location-wise. I'd choose, if it was me, Nijo Castle and Golden Pavilion. That way, you'll have more than an hour (or just about 2hrs) left, so I suggest you include (out-of-the-list) Ryoanji Temple if you're interested in the Zen-style garden. If not, then you can rethink and put Arashiyama on your list: Tenryuji plus inevitable Bamboo Lane plus lookouts for river gorge (plus Ohkochi Villa if possible). And in turn, you'll be pressed for time if you depend solely on public transportation... So in that case/if Arashiyama gets prioritized, you may like to drop either of the two above. A tough decision to make...

If you can text your guide, consult with he or she is the best but I think Kinkakuji Temple, Nijo Castle and making Kiyomizu the last might be the best plan. If you're explained the history, Nijo Castle might look more meaningful. If you can, check the drama SHOGUN. Some scenes are gross but the Warrior days might be really like that.