會 has an entry in 康熙字典 (Kangxi Dictionary):
〔古文〕𣌭㞧𠃐㣛𣥈《唐韻》《集韻》𠀤黃外切,音繪。合也。《易·乾卦》亨者,嘉之會也。《疏》使物嘉美之會聚。《書·禹貢》灉沮會同。《疏》謂二水會合而同。
又《洪範》會其有極。《疏》會,謂集會。《禮·樂記》竹聲濫,濫以立會,會以聚衆。
又《周禮·天官·大宰》大朝覲會同。
又《春官·大宗伯》時見曰會。
又《禮·檀弓》周人作會,而民始疑。《註》會,謂盟也。《左傳·昭三年》有事而會,不協而盟。
又《左傳·宣七年》凡師出與謀曰及,不與謀曰會。
又《集韻》《韻會》𠀤古外切,音儈。與繪通。《書·益稷》日月星辰,山龍華蟲作會。《傳》會,五采也。《釋文》馬、鄭作繪。
又《詩·衞風》會弁如星。《箋》會,謂弁中之縫也。《釋文》會,《說文》作䯤。《周禮·夏官·弁師》王之皮弁會五采。《註》會,作䯤。鄭司農云:謂以五采束髮也。士喪禮曰:檜用組,乃笄。檜,讀與䯤同,書之異耳。
又《周禮·天官·小宰》聽出入以要會。《註》謂計最之簿書,月計曰要,歲計曰會。
又《天官·司會註》會,大計也。
又《周禮·夏官·職方氏》東南曰揚州,其山鎭曰會稽。《註》會稽,在山隂。
又姓。《姓氏急就篇》漢武陽令會栩。
又《集韻》古活切,音括。會撮項椎也。
又《集韻》戸栝切,音活。《莊子·人閒世》會撮指天。向秀讀。
又《韻補》今聲濁,叶泰。古聲淸,叶祭。《卻正·釋譏》三方鼎峙,九有未乂。聖賢拯救之秋,列士樹功之會。
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合计
, and会
was also the word for yearly reviews. This sense of handling the accounting numbers eventually developed into a distinct meaning.