This just isn't true, or at least the evidence doesn't support it at this time. Steppe was practicing stock breeding well before Yamnaya. The PIE package is present in Steppe Eneolithic cultures especially Sredny Stog and Lower Mikhaylovka levels/Kemi Oba. The Iranian_neo had already started to increase at this time well before Yamnaya. What we actually see very early is obvious and significant influence from the Balkans, not the Caucuses, or rather the proposed early evidence of influence from the Caucuses is inferred rather than proven directly. The Corded Ware genotype predates Yamnaya, and it's also different with less CHG/Iranian_Neo. It's missing some sort of Caucasian component that Yamnaya has. So if we agree that Corded Ware types spoke IE, then we have to reject Yamnaya PIE, which I do and I always have. I think Yamnaya was already speaking Indo-Iranian and this is supported by Iranian and North Indian being predominantly descended from Yamnaya, not LBA steppe.
There was no "profound" transformation. An example of that would be farmers moving into the Balkans. On the steppe we see very much the opposite. It’s a vast swath of closely related cultures that show undisputed continuity from the Neolithic to Yamnaya. As the cultures develop you don’t see much difference from the Dnieper to the Volga. There is nothing disruptive happening that would suggest a cultural superimposition like we see in the Balkans, so there’s nothing to make us thing that there must have been a language imposition from the South.
We do have the increase Iranian_neo/CHG admixture, we have Caucuses influenced metallurgical styles emerging in Yamnaya, and what appears to be a freakishly early neolithic on the Volga (6000BC) that most have deduced must have come from the South Caspian via the Caucuses because the eastern steppe is so far away from the Balkans. The evidence consists only of domestic animal bones and maybe egg shaped pots that look South Caspianesque. This deduction is bolstered by the paper showing that farming/stock breeding arrived in the Baltic upon the appearance of CHG/Iranian_neo without Anatolian admixture. That's really about it. We don't have any clear trail of Iranian_neo stock breeders moving onto the steppe and bringing pots with them, much less some sort of neolithic invasion like we see in the Balkans.
I’m not saying something like this couldn’t have happened, I’m saying that the evidence doesn’t show this and we have a very good example of what this would have looked like in the Balkans.
And what then do we do with the Caucasian languages that appear to have extended into Anatolia in the Bronze Age?
Anatolian is the IE language group that most reveals the influence of Caucasian, which is why many prefer a route through the Caucuses to explain the arrival of the Anatolian speakers in Anatolia. But if PIE is in the Caucuses, then where are the Caucasian speakers and how exactly are they influencing early indoeuropean if not from their seat in the Caucuses?
Nalchik and Maykop genomes will likely be telling along with Eneolithic Mikhaylovka and Kemi oba. It’s hard to imagine Caucasian influences not being Maykop mediated, so the interface must be in the Southern Ukraine.
Well, you yourself said it: "The Iranian_neo had already started to increase at this time well before Yamnaya." The Iranian/Caucasian genetic - and certainly cultural - influence in the Pontic-Caspian steppe clearly arrived well before Yamnaya, so I don't really understand much of your comment. That's in fact what I was talking about: not about Yamna, but about the increasing Iranian_Neo affinity before it and right when the transition from Mesolithic to Neolithic ways of life occurs, with CHG/Iranian_Neo already increasing well before Yamna, in Sredny Stog and Khvalynsk/Repin. That's the era where "profound transformations" did happen - and with them or right after them came also an increasing % of Iranian_Neo-like admixture. I don't think that's a coincidence or a small detail.
If it hasn't become clear already, I don't believe that PIE appeared out of thin air as a language and a culture in Yamna, without any links to previous cultures of the same Pontic-Caspian region and, in the longer term, elsewhere.
I also don't believe that there was always just 1 PIE dialect that somehow managed to remain homogeneous and unified for thousands of years and survived all alone to give birth to dozens of language subfamilies. I think there were certainly some missing links, extinct languages, either superseded by non-IE languages and, particularly, replaced by a more prestigious and expansive form of LPIE that was close enough to make linguistic shift very easy and handy for them.
As for the origins of PIE, the hypothesis being discussed now AFAIK is not about its origins in the Caucasus mountains and valleys, but SOUTH of the Caucasus, and as for why there is "no" Caucasian influence actually many linguists have proposed that there is a still noticeable Caucasian influence and mutual connection (loanwords, typological similarities, and so on) between PIE and Caucasian language families, mainly Northwestern Caucasian and Kartvelian. Besides that, there is always the possibility, no, the likelihood that languages have been replaced, moved to other places, become extinct and so on, so the language distribution we see now in the Caucasus does not necessarily - nor even probably - the same linguistic makeup of the Caucasus 7,000 years ago.
Finally, of course we're talking here about some sort of pre-PIE or maybe even "pre-pre-PIE", thus allowing for at least 2,000 years of evolution and heavier influence from other language groups until we have "classic" LPIE, especially after a supposed long-distance migration and extensive mixing with other peoples (EHG-related mainly, but also others). With all that being considered, it wouldn't be any wonder why any "areal features" or loanwords from Neolithic Caucasian - or, more correctly, Transcaucasian languages would be hardly discernible nowadays.
I have no issue with reconciling both an (not "the") origin of the earliest form of PIE, or actually its mother language, south of the Caucasus, but the appearance of the PIE as we know, just before the split of Anatolian, in the north, with steppe cultures. Or even the possibility that those Iranian_Neo people who mixed with others north of the Caucasus simply adopted the local language without imposing their. The genetics don't always correlate perfectly with the dynamics of language, especially languages in varied forms of interaction with each other. But at least in theory, I don't think those linguistic objections you present are very relevant, considering that in Hajji Firuz we're definitely not seeing, not even hypothetically, a PIE speaker, but at best an early ancestor of it who lived in in a linguistic/cultural scenario much different from that of the Yamna period,
As a sidenote, I think CWC only appears north of the steppes when the Yamna were already advanced in its expansion into Sredny Stog territory, so I wouldn't be too certain that the Sredny Stog-like elements that contributed to CWC weren't already linguistically assimilated to a "common LPIE", maybe even some sort of koiné, though in my opinion Sredny Stog may have already spoken a IE language, but not necessarily - and IMO probably not - "the" LPIE that really left living descendants in many Bronze/Iron Age language branches.
