I think that this verb is the only one I've seen in Japanese that has so many definitions. とる and つく have multiple definitions as well (quite a bit IIRC). But not as many as 掛ける. In any case, I am just wondering if anyone knew the reason.
Secondary question, do people tend to use an alternative more often than the 掛ける version? For example: でんわをする instead of でんわを掛ける.
(This is not a full answer but I think too long for a comment):
There appears to be a paper that would cover this more fully.
多義動詞「かける」と「かかる」の意味拡張に関する一考察 - 仙崎 幸子 (A discussion of the expansion of meanings of the polysemous verbs kakeru and kakaru)
However, as far as I can tell this is not anywhere online and you'd have to go fish it out of a library somewhere.
There is some info at this (pdf) study into cases such as why かける was/is used over other possible verbs in cases like 電話をかける. They used corpora such as 青空文庫{あおぞらぶんこ} to look at verb usage for machinery/appliances - there's also some stuff on つける and a comparison chart of which nouns were used with かける・つける.
For 電話を~ they found no usage of anything other than かける, suggesting that this had been the case from pretty much the introduction of the telephone. They considered the possiblity that this was partly because early telephones were primarily wall-mounted (壁掛け{かべかけ}) but found that 受話器{じゅわき}をかける was actually being used to refer to hanging up the phone. Therefore, this meaning probably didn't come from the idea of "hanging" something, like on a wall, as with some uses of かける, but from 声をかける instead.
sethas so many definitions in the English Language). One thing to notice though, is that all the definitions have something in common. They all mean to have an effect or influence on something or someone else, so in your question,電話をするand電話を掛けるhave a slightly different focus.かける/かかるandつく/つける.