會 has an entry in 康熙字典 (Kangxi Dictionary):
The bold text are 5 definitions of 會 that is pronounced as kuài (or ㄎㄨㄞˋ).
According to the 1st of the 5 definitions, 會 with this pronunciation is used equivalently with 䯤 (something like a hairpin), which is pronounced as kuài. This might be from the definition of 會 being the sewings in an ancient Chinese hat (弁) used to constraint hair. Of course, these 2 characters and their meanings could have had their influences on each other.
The 2nd and 3rd of the 5 definitions relate to the definition of accountant. I don't quite see its connection with the hairpin definition, but the usage of this word to represent accounting possibly comes from 會's meaning of gathering together, while its pronunciation succeeded the one for another meaning.
For the name of the mountain and a last name, they might had only a verbal pronunciation kuài beforehand and needed a character, and therefore 會 is used.
So in conclusion, the pronunciation of 會 as kuài might comes from its equivalance with 䯤 (because they look alike), and other meanings with this pronunciation exist only because 會 already has a pronunciation of such, and therefore, derived meanings of 會 takes the sound of kuài (such as accounting), as well as sound of kuài takes the character of 會 (such as 會稽山, the mountain).
(answer based on only Kangxi Dictionary and personal deduction)
会developed from its general meaning of
合, "to assemble".
会计was analogous to
会was also the word for yearly reviews. This sense of handling the accounting numbers eventually developed into a distinct meaning.