Many people put the Fifth Station on their visiting list.
If you go up to the Fifth Station, you could only see part of Mount Fuji like the picture below.
I often compare it to being on the nose. From the nose, you cannot see the whole face, just the forehead.
It's different with European Alps.
In winter, it would be often be covered with snow, and in those cases, the roads would be closed.
Mountain climbers climb up there, for training with snow axes and crampons, by walking up the snowy frozen mountain trails.
If you would like to have a good view of Mount Fuji itself, I would highly recommend the North Shore of Lake Kawaguchi.
Lake Kawaguchi, Hakone and Gotemba are in different prefectures. It is nearly impossible to cover them on the same day with public transportation. Mount Fuji is a huge mountain. Even you can see Mount Fuji in the background of those pictures, Lake Kawaguchi, Hakone and Gotemba are far apart.