Tl;dr: What reasons do we have--besides glottochronology--to think that Proto-Afro-Asiatic is actually 14,000 years old?
So, if you know much about proto-languages, you might know that Proto-Afro-Asiatic (PAA) blows other proto-languages out of the water when it comes to antiquity. The most common figures I see cited in various works espouse a date of 14 or 15,000 years ago. No other proto-language of an uncontroversial language family even comes close to this; I've struggled to find any examples of securely dated proto-languages even half as old. Quite frankly, 14,000 years beggars belief for standard comparative methods, and from what I can gather, there's nothing structurally 'special' about PAA which would make it any more or less innovative than any other language family (the Proto-Semitic triliteral root-and-pattern system, which does structurally foster a particular kind of linguistic conservatism, was not original to PAA, and seems to have come about quite late.)
So I did some digging, and all I could find to back up that "14,000 year" figure was glottochronological estimates. I happen to be a skeptic of glottochronology in general; among other things, it tends to overestimate time depths in cases where vocabulary replacement takes place more rapidly and affects more supposedly 'stable' lexical elements due to periods of intensive interlanguage contact and/or multilingualism (Young Dyirbal being the most infamous example). And there is no doubt that several branches of Afro-Asiatic have substantial foreign components in their phonologies. (For example, recent research has been able to make the case pretty convincingly that PAA was not tonal, and that tone developed independently in Omotic, Cushitic, and Chadic languages as the result of extensive contact with tonal Niger-Congo and Nilo-Saharan languages.) So it seems plausible to me--likely--that the glottochronological figure could be severely exaggerated by mistaking vocabulary replacement due to interlanguage contact with vocabulary replacement due to slow language-internal change. Due to the generally restricted and specialist nature of Afroasiatic studies (and African linguistic studies in general), I do not know of any studies which have investigated loanwords in Afroasiatic languages from neighboring non-AA African languages.
I'm certain I can't be the only person who's ever gone, "14,000 years? Now that can't be right!". So do we have any non-glottochronological reasons to think that PAA is actually as ancient as is so often claimed? And if not, why has this figure has been touted so much in the literature without any qualification or caveat?