This seems like an invitation to nitpick 
What does “it works here” mean? It works in the case of Mount Fuji? Okay, I’ll give you that.
But my guess is that you meant more generally? I could be most generous and assume that you meant it works generally for kanji compounds, but I’m going to be intentionally unfairly restrictive and assume you meant “when talking about mountains in Japan” because then I can justify spending 5 minutes to categorize the top 100 mountains in Japan by whether they use onyomi or kunyomi
Of the top 100 mountains by height, 12 have the suffix 山.
富士山 ふじさん
御嶽山 おんたけさん
別山 べっさん
小蓮華山 これんげさん
白山 はくさん
立山 たてやま
中盛丸山 なかもりまるやま
笹山 ささやま
将棊頭山 しょうぎがしらやま
小太郎山 こたろうやま
赤沢山 あかさわやま
新蛇抜山 しんじゃぬけやま
At least in the case of this sample of mountains, やま is the winner! At the very least, the rule doesn’t help you out with these names.
But you might be thinking, “but that was only 12, didn’t you say that this was a list of 100 mountains?”
Well, the overwhelmingly most common “mount” suffix in Japanese (for tall mountains anyway) appears to be 岳, which has the onyomi of がく the kunyomi of たけ.
More than 80 mountains on that list use that and they all either have the reading of たけ or the rendaku’d だけ.
So kunyomi wins by a landslide (pun intended) for mountains in Japan! Nit thoroughly picked.
EDIT: If you keep going beyond 100, やま gets more common.