Am I right to rephrase your question as:
why かぎのあるヱ transcribes the character by the shape,
while ゐどのヰ, along with all the other ones, transcribes the character by the sound?
Today, standard Japanese doesn't distinguish ゐ/い ゑ/え を/お by their sounds.
This means that virtually any contemporary Japanese would write "い"
when requested to write "ゐどのヰ" in their daily conversations.
Historically, Japan joined Convention on International Civil Aviation,
which required standardization of radiotelephony alphabet, in 1953.
This suggests that ゐ was pronounced still differently than い in
daily (or even official) conversations around 1950's, while ゑ was already pronounced the same way as え.
("現代かなづかい", which ordered "ゑ" be written as "え" etc. in public, was enacted in 1946.)
I didn't notice that my four grandparents, all born around 1930s in Kyushu,
distinguished ゐど from いど or ゑびす from えびす. But I cannot tell for sure, because their pronunciation
was always shifted somewhat from our Tokyo dialect.
However I found this from twitter:
Emperor Hirohito was born in 1901 in Tokyo, and perhaps would represent the
pronunciation of then Japanese upper class speech at the time. However, his
son, Emperor Akihito, never distinguishes them in public
speech. He was born in 1933.
So I think you are right in saying that "there's more to it", because
all this indicates that Showa was a subtle but major phonological
transition period. But this may require a serious linguist to analyse,
and I refrain from talking about it other than from my personal experience.
(In kindergarten I used to believe the "correct" pronunciation of を
was "wo", and was not the same as お, because they are spelled differently. Some comic
books even used them in proper names, as in アキヲ.
But since my friends and teachers never spoken that way
I gave up the idea.)