Agreed.

There aren’t many places in the Hakone area for such a shot. Fuji San is only visible from certain locations and the best known spots are from a Lake or a cable car. If you were to go “off piste” and try some creative hiking, you might possibly find a rogue cherry tree somewhere to provide a frame for the picture, (hint- it isn’t on the mountainside around the Green Plaza Hotel/Ubako Cable Car Station - the onsen bath at this hotel is a celebrated Fuji viewing place.) One of the few well known places for Sakura and Mt Fuji would be the Onshi Hakone Park - which is described in this Japanese website
https://hanami.walkerplus.com/detail/ar0314e60506/
You can feed it into a translation tool for more information.
The map reference would be “ 171 Motohakone, Hakone, Ashigarashimo-gun, Kanagawa 250-0522” or 神奈川県足柄下郡箱根町元箱根171 Copy and paste either into Google Maps for a precise location.

The second link I added (post #2 ) is what will be the ongoing Sakura prediction for the Northern shore of Lake Kawaguchi. That’s where most of the famous Fuji and cherry blossom photos are taken from.

The title and website from the original post is vague about Fuji-Hakone-Izu National Park not being one seamless area. To make sure I’m working on the same idea of what the Fuji-Hakone-Izu National Park area actually means, here is a map.
All the photos showing Sakura and Fuji in that introductory website are taken from what is labelled as the Mt Fuji Area.