Regarding the overall itinerary it may be fine for you if you are quite eclectic, have a generally fast travel pace and really don't see yourself going to Japan ever again, or at least in the near future. In principle I agree that running around too much is too hectic for a lot of people.
Takayama is worth a visit, but unless you are making a whole side trip and including other places like Shirakawago and Kanazawa, you could drop it. Instead, you can still stay in Tokyo, and add a trip to Nikko. Or, add more time for Hiroshima.
For Hiroshima I just hope you give the city its due and not just see the Peace Park and leave the city, as so many stampeding from the station do. There is much more than that, so try to add in something more, like the Shukkeien Garden, Mitakidera Temple, or eat some okonomiyaki at Okonomi-Mura. After the Peace Park you'll likely need to decompress some anyway. There is more than enough for 2 days to see both Hiroshima City and Miyajima, but you can do both in a day with an early morning start. The main sights at Miyajima are the Itskushima Shrine, Daishoin Temple, and Mt Misen. If the weather is bad, then you can skip Misen, though October is one of the best months for good weather.
https://goo.gl/fAz4axI highly recommend seeing Koyasan, and especially Okunoin. There is simply no other place in Japan like it, and it is a truly surreal place. But if you want to see it at night as well (a very different experience), I implore you to choose a temple that has no evening curfew. You can cover all the main sights there if you are there for 24 hours.
https://youtu.be/uquvCbWANBoFor Kyoto, obviously the temples and shrines are the main draw. If you detest them, you have probably picked the wrong city. While you can get "templed out" just going from one to the next, the key is to add enough variety so everything in your head is not just a blur by the end of the day. There are enough places you can include to do this quite easily, and they are not at all dull or below average places. A few are Katsura Imperial Villa, the Toei Eigamura, Heian Shrine Garden, Murin-an, Sagano railway + Hozugawa river cruise, the railway museum, NIjo Castle, Nishiki Market, Iwatayama Monkey Park, Shoseien Garden, and many others. I'm not saying to add all these to your plans, but you can have a variety of places, even in Kyoto.
https://goo.gl/knQtMgOne more option you have to give yourself more breathing space is to skip the train all the way back from Hiroshima to Tokyo, and fly. You can get a dirt cheap airfare for example on ANA's Experience Japan Fare (11,000 yen per person), and fly back from Hiroshima to Tokyo on your last day. It can save several hours and may work better for you, unless you have big shopping plans before you leave the country. I would keep Himeji though and see it on your way down from Osaka. Your initial 3½ days in Tokyo is a pretty decent amount of time to get your toe in the water.
One thing you never mentioned is visiting a hot spring somewhere. I highly recommend you visit at least one during your visit. They are all over the country, so you should be able to fit one in, and it is a real Japanese experience.