First of all, this is not my query but that of Mitch.
The IPA system has two great advantages: the symbols are standardized, and everyone may find the acoustic value of a symbol in his own language.
Not using the IPA put one in the obligation of using symbols which value is not easy to find and, as Katalogon well underlined, of finding equivalents in various languages which the readers of the same language of the writer does not know.
And indeed Allen appears to be using:
— for IPA [e], the close mid front: an e with a point under it and
— for IPA [ε], the open mid front: an e with sort of an inverted comma under it
— for IPA [e̞], the mid front: by using his chart on page 62 as Katalogon did we may suppose that Allen uses a plain e.
Page 63 he writes:
“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. 3” The view that they were of a specially close mid quality, i.e. [~], [~], as in French gai, beau, is probably
mistaken (cf. pp. 72, 89 f.).
Note 3 : “The vowel of English pot is decidedly less accurate, being fully open rather than mid.”
In the article about the mid front vowel [e̞] here:
https://en.wikipedia.org/wiki/Mid_front_unrounded_vowel
the example given for the Received Pronunciation is let [le̞t] and there exists no mid front vowel [e̞] in the French phonology.
Then page 64, after a paragraph about the similarities with Latin î, Allen concludes by:
“Thus the E in Greek words of this type may, even in Attic, stand for a specially close variety of [e] which would then be particularly near to the Latin î. 4”
Finally, in The summary of Recommended Pronunciations page 177 one only finds:
ε As in English pet 63f.
Now the problem is that the pronunciation of pet is:
/pet/ in the Cambridge Dictionary:
https://dictionary.cambridge.org/dictionary/english/pet
/pet/ in the Oxford Learner’s Dictionary:
https://www.oxfordlearnersdictionaries.com/definition/english/pet_1?q=pet
but /pɛt/, [pʰɛt], [pʰɛʔt] in Wiktionary:
https://en.wiktionary.org/wiki/pet#Pronunciation
So by not using the IPA, you cannot know precisely what one is saying.
The Cambridge Grammar of Classical Greek has the great advantage of using the IPA system: at least you know what they mean.
It would be very interesting to know the reasons of the choice of the writers of the CGCG of the pronunciation of epsilon as the close mid front vowel IPA [e]. As Mitch has noted, they agree with Threattle (who page 137 define epsilon by the IPA [e]).
Of course I am an absolute layman here and for sure I don’t understand everything but, for what I have read, the reasons of Allen appear to be neither very numerous nor very strong.
You are all very experts on Ancient Greek: how do you analyze the work of Threattle about epsilon being IPA [e]?