Kyoto has gotten a huge amount of bad press these days because of the huge crowds that are now guaranteed at all of the "must see" places (making them "must avoid no matter what" places as far as I'm concerned), but in my own personal experience, there are dozens of places in and around Kyoto that are not at all crowded, but are as good as, or better than, the "must see" places in some of the less famous cities. My recommendation would be to spend more time in Kyoto as a home base, and go to some of these sublime but less crowded sites (skip the "must see" places altogether, is my strong suggestion). Examples might be Mimurotoji and Manpukuji in Uji (both of which I personally find much more enjoyable than the "must-see" Byodoin there), various temples in Ohara (including but not limited to Sanzenin), and Yoshiminedera (I hate to even mention this one because I think it is an absolute gem, and it would be a pity if it got "discovered" and a lot of people started going there). These are just a few examples; if you do a little research you can find a LOT more, some of them being closer to central Kyoto (but you said you don't mind a little travel, and most of these places are heavy on the "nature" aspects; Kurama is another one if you want a nice hike). The point is, why flit around to a bunch of far-flung places if you can comfortably base yourself in a dynamic, multi-faceted (and quintessentially "Japanese") city with loads of great lodging and dining options and make leisurely day trips? Note that the places I mentioned are ones that Japanese people are likely to go to (moreso than the "Top Ten" places, which they have probably already been to and won't want to visit anyway because they don't like huge throngs of foreign tourists), so they will have a bit more of a sense of "real Japan" than the places with endless busloads of visitors. Also, all of them have wonderful outdoor settings and/or fine gardens, along with oodles of history.
Gunma is a good idea. Or possibly Nagano prefecture (Togakushi is indeed well worth visiting!), which also has hot springs and some very good scenery. (The Koumi line provides an outstanding segment for a recreational train ride, and you can't get any more "middle of nowhere" than that.) I would skip Kanazawa simply because aside from Kenrokuen I don't think it has much that would be particularly unique, for a first-time visitor. (It is a very nice city, but you have limited time.) If you want to see a fabulous landscape garden, make a day trip from Kyoto to Ritsurin Koen in Takamatsu, which will also give you a very enjoyable train ride. It will likely be much less crowded than Kenrokuen.
Miyajima and Hakone are two of the worst places you can go if you want to avoid crowds, although you didn't specifically mention that as a priority. But I think you can find equally nice places if you want to experience the "real" Japan. For views of Mt. Fuji, how about Minobu-san (a gem, in my book), Shimizu, or the Miura Peninsula? All of these are easy day trips from Tokyo and much more "real Japan" than the places where tourists outnumber locals by a ratio of at least 10 to 1. But you need a bit of a spirit of adventure and a willingness to do your homework to visit such places.
Finally, especially if you have a rail pass, Tohoku offers numerous possibilities for splendid day trips out of Tokyo (maintaining the efficiency and comfort of a stable home base without changing hotels), or you can possibly spend a night or two there. This could be an alternative to Gunma if you want a nice hot spring.