Nokogiriyama has a lot of walking. If you drive to the cable car (ropeway) and take it to the top, you get a good view over Kanaya but you're not there yet. You have to hike down to a gravel parking lot, pay the entry fee, and begin the steps up the mountain to the quarry and overlooks. You could also drive to this parking lot but there's not much there.
At the other end of the mountain on the south side, you could drive up to the parking lot just below the stone Daibutsu. You have some steps to go up but not too many.
Kamakura has a lot of sights but they seem to be in patches of woods in a largely urban area. Parking can be difficult. Most people walk the trails between the temples, but you could probably find somewhere to park around the train station, the Daibutsu, and Enoshima.
Nikko would be a good place to have a car. To see the temples though, you have to walk uphill. The car would get you up around Lake Chuzenji with stops along the way such as Akechidaira for the Kegon Falls overlook. A car will get you to the northwest part of the lake area to some waterfalls, marshes, and an upper hot spring area. On the way to Nikko Edomura in the lower part of Nikko town is a shop called Oshinko-mura on the right side of the road. It has hundreds of different kinds of pickles.
On the east side of Utsunomiya (downhill from Nikko), is the pottery town of Mashiko, if that interests you.https://www.japan-guide.com/chottozeitaku/161125.html
In the town of Sakura, heading out toward Narita Airport, is the National Museum of Japanese History. It's in the grounds of the former Sakura Castle of which little remains.https://www.japan-guide.com/e/e6407.html
In kasukabe, Saitama is the Metropolitan Area Outer Underground Discharge Channel , an interesting tour of a huge engineering project built to divert flood waters from Tokyo. You have to have a Japanese speaker with you for the tour though.http://www.ktr.mlit.go.jp/edogawa/gaikaku/english/
Again, I would advise that you aim for the edges of the Kanto Plain, as it's almost totally urbanized within with corresponding traffic. The Kanto from Takao-san (west side of the plain):https://www.flickr.com/photos/anaguma/5238512655
You could also base yourself just outside the Kanto (43,000,000 people, 1/3 of the total population of Japan, and nearly as many cars) in someplace like Kawaguchi-ko by Fuji-san or Takasaki on the NW, and you wouldn't be bogged down quite as much as being in the middle of the largest conurbation on earth. Tokyo and the surrounding prefectures of the Kanto bog you down and suck up time getting between two points. If you're planning on hitting up Kabukicho every night, then Shinjuku is a good location (I wonder what happened to my bottle of whiskey I left in a snack bar there from my student days). It's not a good base to see the surrounding areas.