My own mother has stipulated that she wants to see Mount Fuji on her next trip (next Spring), so I’ve come up with a flexible itinerary that allows for her to stop at Shizuoka as she travels between Nagoya and Tokyo. From Shizuoka she can make her way to Fujinomiya station, nearby is the Fuji World Heritage Centre (there’s a basic café there). If the day is clement, there is the option of an hourly bus that goes from Fujinomiya to Shiraito Falls (ride takes about half an hour)
If the weather is clearly unfavourable from the outset, she can ditch the Fujinomiya detour and continue from Shizuoka to Tokyo without wasting any time. I give her a 30% chance of success.

Plan B is to periodically try her luck by going up tall buildings in Tokyo.

The flexibility is the key point. Since one is unlikely to see Mount Fuji on any given day, I cannot see the point of arranging a tour. Yes, some people are lucky, but that’s not a suitable basis for a responsible plan (or advice). The chance of seeing Fuji San is increased in the early morning (before 9am) and just before sunset, so it might be worth hanging around if you make it to a vantage point and have been disappointed.