A lot of sources I've seen online (Wiki as an example, not that I trust them) claim that the short vowels were closer than the open vowels, but you may be right.
I'm presuming that the Wiki article intends to be citing that entire section from Allen, and, if that is so, they are misrepresenting how he describes the vowels. To quote him "There is no reason to think that the sounds represented by these letters were ever other than short mid vowels, front and back respectively, i.e. rather like the vowels of English pet and German Gott. The view that they were of a specially close midquality...as in French gai, beau, is probably mistaken." Allen says that ε is midway between ει and η later on and this is because sometimes ε lengthens to ει and other times to η. I think that this may not be a very strong argument, though, because he doesn't really take into consideration that the instances when the two different lengthenings occur are different.
I don't entirely agree. I mean we should use them as a model, but I don't think we should slavishly adhere to their (reconstructed) phonology if there's an easier alternative. In the case of Latin, for example, I don't see the point in pronouncing "-um" with a nasalized vowel when a simple [m] will do. In the case of Greek, I think it's easier to turn the aspirated plosives into fricatives than to maintain a three-series plosive distinction.
I was more just (jokingly) objecting to the one city comment since what beginners learn is Attic Greek, which is the language of Athens, so it doesn't really make sense to care how things are pronounced in other cities. I wasn't objecting to emending the pronunciation to something easier for an English speaker, I certainly do, too, and it doesn't really make much of a difference since there's no one you're going to be talking to anyway.
I've seen a lot of disagreement over its phonetic value, and [z] is simpler.
No one seriously argues for a [dz] pronunciation anymore, this is a long dead argument (though, as a side note, it is likely that what became zeta grew out of a convergence of [zd] and [dz] into [zd], this change [if [dz] did in fact still exist in the major dialects] happened almost immediately after the alphabet was adopted - and there is evidence that several non-major literary dialects had a phoneme [dz] or [ts] that contrasted with [zd]). Granted that [z] is certainly easier to say than [zd], and, if you're interested, zeta was pronounced [z:] intervocalically in the Roman period, which will help with Greek poetry as zeta is counted as two consonants.