You don't need to join a tour. However, I would get a travel company to organise the Rail card and your itinerary. Takes a lot of stress out of it and you get to go to places that interest you, not other people.
We used Audley Travel and we did stay at a Ryokan. It was at Tsumago. Audley put a fabulous tour together for us and it was truly memorable. I too was concerned about the Ryokan but it was no problem. The hot tub, onsen, could be booked and the loos/showers were available all the time. They were in a different area of the Ryokan, but it was a small building. As you booked the onsen for your personal use, there was no-one with you. The Japanese tend to go in as a bigger group, although in our Ryokan, not that many could get in the onsen. These Ryokans are popular. You also have traditional food but in ours, the host knew western guests wanted western breakfasts too. I think you could survive being a vegetarian! The tatami mats are fine and I would have thought they were good for backs.
On our holiday, we mostly stayed in hotels. However, the Ryokan hotel was a great experience for two nights and it was in the old post route which offers some great walking. It is also quiet, unlike nearly everywhere else. The one we stayed at was not especially luxurious, but it was welcoming, charming and comfortable with superb food. Very high ratings too.
The Japanese are unfailingly polite and no-one would comment about you. They just don't do that. There are lots of onsen, so a Ryokan that has suitable facilities is important and companies like Audley can advise. If you want a more private experience, you can have that.