I’ve found the imabi website some time ago and I can recommend the first two chapters about pronunciation in this case. The author seems to be somehow a person with knowledge in linguistics, so I guess it won’t be all that wrong.
There are actually words like 姉さん(ねえさん) where you stretch without the い. To sum it up, as far as I understood imabi. lengthening the お with おう and the え with えい seems to mostly appear in on’yomis (hence orignally chinese words). My personal guess is that a long time ago the japanes people maybe tried to pronounce the words more closely to the original chinese pronunciation, but over the time the sounds just became long お and え. Unfortunately, I have no proof or historical hints. So there is a good chance that my theory is plainly wrong
However, what ever the reason might be, it is indeed an interesting question.
Regarding the えいえ: If you would not pronounce the い it would result in a triple え. I think it would just be to confusing and impractical if the speakers of a language had to distinguish between three different syllable lengths, so pronouncing the い more clearly has established. Especially since the meaning of many, many japanese words already depends on whether a vowel is long or short (ばあさん - ばさん, 有名(ゆうめい) - 夢(ゆめ)). But I’m not sure either.
Anyway, I agree with the previous answers: The “correct” pronunciation of えい seems to be ええ, and, as seanblue already mentioned, don’t try to imitate songs or so. For example, in German you write very often "silent e"s, like in “haben” (have), but never speak them. Doing so sound very strange and unnatural, still in songs you sometimes here it because of various musical reasons (beat, metric, etc.).
ps: this is my first post, too yay