Hi, moon369,will push it if it means extraordinary food and experience but also looking for authenticity and value.
This is very difficult question because it depends on individual taste. Therefore, the following is just my opinion and please do not get me wrong.
If you look for "authenticity", I think you should find the ryotei (Japanese-style fine restaurants) from Kyoto cuisine association (京都料理組合).http://www.kyo-ryori.com/50on_index.html
Obviously there are many other fine Japanese restaurants in Kyoto, as you mentioned, however, I think you should better experience the "standard" restaurant first. You can check each restaurants about their features and price ranges. You will find there are very expensive ones (for example, Kitcho Kyoto-京都吉兆, starting from 36750 yen for lunch) and rather affordable ones (for example, Imobou Hiranoya Honke-いもぼう平野屋本家, starting from 3150 yen).
I have been to only several member restaurants. From my few experiences, I recommend Manshige (Manshige) in Nishijin. This ryotei is not in the central Kyoto. In general, the fare is cheaper in the off-central region. Manshige has about 20 private rooms in the traditional Sukiya or Mingei styles. The guests will be introduced into one of these rooms and they will be served authentic traditional cuisine in the private room. There are no other guests which will come after you, and you will enjoy your lunch or dinner without taking care of time. I recommend to reserve lunch at your first visit, since the lunch fare is cheaper than dinner and you can go back your hotel easier by public transportation. The foods in Manshige is not so novel or original: they are traditional or old-fashioned Japanese foods and might be somewhat strongly seasoned, especially for their “spécialité de la maison”,
Tai no Aradaki (simmered sea bream). Today, many Japanese tend to prefer less seasoned cuisine (usuaji-薄味), especially in the sophisticated Japanese style restaurants. Therefore, Manshige is not so popular among Japanese foodies, however I believe it is accepted as a tasty casual ryotei among people living in Kyoto. I visited Manshige several times when I was living in Kyoto, and I was very satisfied every time. It is not so cheap (also it is not so extraordinarily expensive), however, I always felt it is more than valuable for the fare.
The website is in Japanese only, however you can see photographs of their rooms and cuisine.
You might be able to reserve via e-mail. firstname.lastname@example.org