> But the general point is that the area with the widest range
> of languages in terms of number overall, and in terms of
> phonemic diversity, are in Africa. Which should not be much
> of a surprise since the location of origin for languages in
> general have greater diversity due to divergence over time
> and lack of sustained interaction between dialect groups.
Actually no, that is not an expected result. Genetically, Africa is linguistically relatively uniform, with most languages of the continent belonging to only two families: Afrasian and Niger-Congo. These have come about relatively late in the global scheme of things. Only sporadic islands remain that
could represent original variety (the best-known cases being Hadza, Sandawe, and the recently identified Bangime).
There are the numerous "Khoisan" groups too, which, contra Greenberg, have not been satisfactorily related to each other, but they have definite typological commonalities, e.g. the presence of clicks (which have also been introduced to their Bantu neighbors).
> Which leads to the secondary areas of great diversity in
> phonemes and languages in general, New Guinea and
> (pre-contact) Australia.
Not quite right either. New Guinea has vast genetic diversity of languages, but the area also has a high tendency towards rather minimal phoneme inventories for some reason, which limits the possible variation. Australia is also home to a large number of language families, but is incredibly homogenous in terms of
phoneme variety in particular.
A much hotter bed of linguistic variety is found in NW America (particularly California), probably much for the same ultimate reasons as at the Caucasus: the mountaneous topography has allowed a large variety of ethnic groups to coexist. The area of the eastern Himalayas also shows a similar effect, though this must be secondary, as most languages in here are known to be Sino-Tibetan or Austronesian.
> Now to get super-speculative. The Caucasus was also a region
> where Homo sapiens neandertalis survived the longest and
> interbred with Homo sapiens sapiens. If they had language,
> some of this language may have contributed to the diversity
> of languages in the Caucasus region both in terms of
> phonemic diversity and general language diversity.
The languages of the Caucasus are probably mainly relicts from various prehistoric expansions thruout the millennia (much as is the case of Ossetic and Armenian), not ultimately indigenous. Also, recent newcomers aside, the area's language diversity here too reduces to just 2-3 language families, so much of the variety is again clearly secondary.
Altogether: synchronic phoneme variety in an area does not correlate particularly strongly to the genetic variety of language lineages. If the conditions are right, unrelated languages can apparently relatively quickly (over a millennia or three) evolve towards a similar typology, while related languages can just as well typologically diverge to the point of seeming unrelated at first glance.
_j.