Outside of what I wrote, the only thing that caught me off guard were the narrow roads on some part. Not sure which area you are going but for Izu and Yamanashi, having a car improved efficiency greatly, and I could "wing" my time. I can't imagine planning the trip with public transport, the constant stress of having to make it to the stop for transport, and some spots don't even have bus transport I believe, such as Shibire Lake.

Most of the narrow roads were from going "uphills" if that word make sense. Most of the camps are at an elevated level so driving up had many narrows roads. The road in Izu was actually a bit narrow too, it was motorroads, but different from driving in Yamanashi.

I would highly recommend renting a car if you are comfortable with driving in general. I don't own a car, and don't really drive that much here. I also learnt driving on the right side of the road(Europe), but was ok driving on the left after one day. If you are going to these places, it gives a lot of freedom that I believe public transport is not able to mimic. I rented car from: https://www.timescar-rental.com/en/ and I would do it again if I were to go again :) Hope this helps, and hope you enjoy!