While I agree with the principle that there's no reason to assume sudden change in the Caucasus without evidence, I think your argument there's a little weak. After all, Mayan and Na-Dene aren't the only families to show considerable typological uniformity after thousands of years of divergence. Another classic example is Romance, where the languages have maintained a great deal of -in some cases quite detailed - similarity despite 1500 years of divergence.
The problem is, Romance DIDN'T maintain that typological similarity. It innovated it, across the whole of the continent at once. No sane reconstruction of Proto-Romance would look anything like attested (even vernacular) Latin!
But of course, there's no specific reason to think that that happened in the Caucasus. The typological divergence between Caucasian families discourages the idea of a recent sprachbund, while Caucasoid features in PIE encourage the idea of at least some areal similarities, if not genetic connection, in the distant past.
Howl: in the specific case of the Hittites, yes, we know that there were ethnic distinctions in the polity. Most of the core area ruled by the Hittites had, until a short time before, been ruled by the Hattians - hence the name of the Hittites, which comes from foreign names for their country as "land of the Hattians" - who were a non-IE group. The Hattians were conquered by the Hittites. We know there was preservation of the Hattian religion and (at least in liturgical contexts) language well into the Hittite era. So it's not at all surprising that if you randomly pick five (iirc) people from the Hittite era who weren't from the ruling class, you wouldn't get any obvious Hittite genes.
If the Hittite ruling class don't look IE, that's more surprising (as long as they aren't very late, since the Hittite ruling class was iirc in turn Hurrianised later on). What would be more surprising still would be early Hittite samples from Nesh, or samples from other Anatolian groups where there isn't such a clear multiethnicity.
The potentially disruptive news, btw, is that some people believe that there are seemingly Anatolian names found in Eblaite records of the inhabitants of "Armi", an unknown, presumably highland Anatolian, polity. This would be really important, because those names would be found in Anatolia at a time contemporary to Yamnaya culture north of the steppe, and apparently not looking like a distinct recent migrant class. Given the lack of any obvious migration from the steppe before then, and the lack of anything obviously migrationy about the alleged Anatolians in Armi, that would not only prove a much earlier separation between PIE and PA (effectively we'd be firmly in Indo-Hittite territory, conceptually), but would also make the prehistory of the language really puzzling. Given the very low levels of steppe genes in anatolia, and the high level of non-steppe genes on the steppe, it would suggest that Indo-Hittite developed south of the caucasus and then migrated north, either via the caucasian piedmont*, or more radically through europe (radical because this could even put LBK back on the table as Indo-Hittite**).
*culturally this has always been suspected - there seems, for instance, to be a development from mesopotamian ziggurats to armenian mound tombs to steppe kurgans, as well as some transmission of metallurgical culture. But there's no obvious genetic pathway for this (it turns out Maykop, the obvious link, wasn't that genetically related).
**the radical theory there would be that mesolithic europe was indeed Indo-Hittite, that Anatolian is a back-migration from Europe, and that an influx of farmer women brough the language to the steppe. However, since the cultural and genetic ties to europe are even weaker than those to the caucasus, this seems less intuitively likely.
But the river tripped on her by and by, lapping
as though her heart was brook: Why, why, why! Weh, O weh
I'se so silly to be flowing but I no canna stay!