In my opinion, remembering all together as "one name", because there is no specific rule, they have been called like that for a very long time. Besides there is one more case which 山 is read ZAN. (Ex. 有珠山/うすざん、大雪山だいせつざん). I hope this was helpful.

Regarding your study of Etymology, the pronunciation came from what local people call it since prehistory, not relates to any other things, and after 漢字 came from China our ancestor assigned three pronunciations to 山 as Ryoko-san explained.