This etymology is found only here. From ἄγω, the action noun ἀγή is indeed attested epigraphically, but is very rare, so maybe ἀγή here is only a mere abstract construction from ἄγω for the sake of derivation. The meaning would be that youth is the result of "driving, leading" toward adulthood – unless it must be understood as "gathering" of young people of the same generation (the epheboi in Athens). The formal aspect is difficult because of the formulation: what is added should be the [m], which is not mentioned. If it is not corrupt, the formulation means that from ἀγή one derived *ἀγκή (through addition of [k]), and as is known, ‹γ› before a velar stop is the regular spelling for a nasal (ἐμβάλλω / ἐγκαλέω), so that this *ἀγκή is equivalent to [aNkē], which can stand for [ankē] of [amkē] and can legitimately, if submitted to metathesis, become [akmē]. This is an instance of confusion between the graphic level and the phonological level. Ancient Greek etymology has many instances of such a confusion, since the Greeks were reasoning with letters, not with sounds, and for them sound and letter were one and the same thing. Here the assumption is that adding a velar to [ag] (ἄγω) which has a real velar, graphic and phonological, will transform it into ἀγκ-, which has a graphic velar but a phonological nasal, and then metathesis will turn it into ἀκμή, which has a graphic and phonological nasal. The phonological level is required as an intermediate step between two incompatible spellings, with the letters Γ (ἄγω) and Μ (ἀκμή).