I love Japan's trains and other public transport... but I think that a car opens up a bunch of possibilities around Mt. Fuji and Yamanashi.

Most people who rely on public transport from Tokyo to Fuji make a return trip to Hakone or Lake Kawaguchi. It's a fun trip and I have nothing against it.

But if you rent a car from Hakone and drive around the North of Mt. Fuji, you can see much more "nature" in the less developed areas around Lake Motosu, for example. (Motosu is the lake on the back of the 5000yen note). There are some walking trails into the hills to the North of Mt. Fuji that you can't easily reach without a car. The views of Mt. Fuji from those hills are spectacular.

You can also take roads that go deep into the "terrifying" forested areas around the foot of Mt. Fuji (the "Aokigahara") and then up into the "highlands" - a plateau of golf courses and dairy farms.

If you want to extend the road trip to other parts of Yamanashi, I suggest driving to the North of Yamanashi - up to the Yatsugatake mountain range. Up there you can enjoy some fabulous walks, see the "Iron Road" that the warlord Takeda Shingen built to enable his army to get around the province, enjoy some ice cream in Kiyosato. Frankly, the panoramas from the "skyline" toll roads up above Kiyosato are more impressive than those you can enjoy immediately around Mt. Fuji.

And if you really enjoy the driving, you could continue from Yatsugatake into Nagano prefecture.... see Lake Suwa... visit more onsen... end up in Matsumoto. (And yes, you would probably end up paying some hefty expressway tolls if you wanted to get there quickly).

In addition to the grand views, there are also a couple of scenic valleys in Yamanashi - You could see one (The Ashigawa valley) if you drive from Lake Shoji to Ichikawa Daimon. Alternatively, from Lake Motosu, you can take a road down towards Shimobe Onsen which is very picturesque. There's also one very famous valley called "Shousenkyo" to the North of Kofu. (But this one gets very crowded in the holidays).

Of those valleys, I think only Shousenkyo is readily accessible by public transportation.

So -- All in all, I think that there's every reason to rent a car if you want to explore the Fuji Five Lakes region and other areas of Yamanashi such as the Yatsugatake mountain range.

OTOH, if you just want to visit Hakone and then go to the fifth station of Mt. Fuji and climb the mountain, you can do all that by public transport.