This is one thing that is bothering me in this whole "Transcaucasia vs. Steppes" discussion. I mean, if the bulk of the CHG in the Pontic-Caspian steppe populations is very ancient, having being absorbed in high proportions before 5000 BC (the lack of ANF in the steppes also indicate that early introgression), then it is, from a linguistic point of view, very hard to accept the possibility that Early PIE was a "CHG language" from the southern slopes of the Caucasus, because that would imply that Anatolian IE (or also possibly, according to this study, even much less diverged languages like Armenian and especially Greek) and the Pontic-Caspian steppe IE would've split in the 6th millennium BC. Either scientists assumed a much faster pace of linguistic evolution, or it is just impossible that the initial split of PIE happened so early.
Now, if we could demonstrate that there was a significant amount of extra CHG in the Chalcolithic Pontic-Caspian region, that could suggest a new demographic and presumably linguistic layer onto the older ethnic makeup.
As for the lack of EHG in those few "Hittite" (or at least "near to the Hittite") samples, I was thinking (okay, speculating) a bit about it from the assumption (pretty much mainstream among linguists) that Anatolian IE split much earlier than the others and probably in a very different historic context (certainly not the mobile horse-driven pastoralism of Yamnaya and descendants). It also seems from this paper that the North Caucasus, right next to the steppes, had a very different genetic structure with a much higher CHG and ANF, so I'd think it is plausible that the southernmost portions of Sredny-Stog and/or Khvalynsk in direct contact with the North Caucasus could have some substructure in a transition zone to the steppes. We know now that Maykop had EHG, which is not found further to the south, so there was some cline. If the ancestors Hittites came from this region and expanded more or less in the fashion of later IE branches into regions that were already very populated (like Greece and South Asia), then they could've migrated south becoming a relevant and dominant minority with an increasingly diluted DNA makeup, and given their very early separation from the steppe or North Caucasus populations it's probable that by the time they established in former Hattic-speaking lands to form their kingdom and empire their EHG portion was just too small to make a significant presence in the genomic makeup of the region's average inhabitant. Mere genetic drift and regional substructure could make EHG virtually invisible after a few centuries unless we had many more samples. Doesn't this type of thing happen when migrations were not that powerful to trigger an appreciable population replacement? Just playing a bit with this speculation, let's imagine this totally hypothetical (and admittedly baseless for now) scenario:
From CHG-enriched southern steppes just north of the North Caucasus (South_Steppe): 40% EHG, 55% CHG, 5% ANF
Admixture with (north or south?) Caucasians during the Proto-Anatolian phase: 25% South_Steppe + 75% Local Caucasians (5% EHG, 70% CHG, 25% ANF) >>> 13% EHG, 66% CHG, 8% ANF.
Admixture with North-Central Anatolians during the Hittite phase: 20% Proto-Anatolian + 80% Hatti and other natives (0% EHG, 30% CHG, 70% ANF) >>> 2.5% EHG, 37% CHG, 60.5% (2.5% EHG - and that's assuming that the Caucasians already had some EHG even before Maykop and that the Proto-Anatolians still made a reasonable demographic impact of ~25% and the Hittites one of ~20%, not too shabby)
I'm just entertaining all the possibilities, especially considering that apparently the CHG component in the steppes is MUCH older than even the 1st Indo-European split around ~4000 BC, and not just some Late Neolithic/Chalcolithic influx bringing not just a new (presumably more advanced) people but potentially a new language family.