Such a proposal has its appeal in terms of the culture from what we know of the archaeology, and it
may turn out that the genetics would support it, but I'm not aware of anything published by a linguist that fleshes it out.
The Anatolian languages staying in Anatolia
would explain the problems that the Anatolian languages present. I know you're aware of all of the following, but for those who aren't:
From Mallory: "Twenty-first century clouds over the Indo-European homelands."
http://jolr.ru/files/(112)jlr2013-9(145-154).pdf
"The essetial argument as it is normally presented is that Anatolian lacks a considerable number of features that would characterize Brugmanian Proto-Indo-European (aorist, perfect, subjective, optative, etc.; Fortsom 2004, 155) and, therefore, its links with an earlier continuum must have been severed before Proto-Indo-European (or the rest of the Indo-Europen languages) developed in common. This can essentially be explained in one of two ways:
l. The ancestors of the Anatolian languages migrated from the homeland of the proto language before it developed common Indo-European features. In this model Anatolian would have preserved an archaic structure while the ancestors of the other Indo-European still remained together and evolved later stages of Indo-European.
2.The ancestors of the Indo-European languages migrated from the homeland of the proto-language. Here it is proto-Indo European that moves off to innovate, while presumably Anatolian was left in the homelans to preserve its archaisms."
Number 1 is the Baltic route model to which Anthony and Ringe adhere. I'm not sure they're right. I've combed through "The Horse, The Wheel and Language" and his claim is based on the archaeologically attested movement of what he claims were Indo-European people very early down along the narrow, western coastal strip of the Black Sea. I couldn't find any place where he shows further movement into Anatolia.
Number 2 is close to the model that Renfrew now seems to be floating? If he believes that, retired emeritus professor or not, I wish he, or someone else for that matter would publish a paper fleshing it out. Otherwise it's difficult to give it much weight. I tried to read Grigoriev's tome on the archeology, but I have to confess that I stopped after awhile. The length was daunting, and turgid doesn't begin to describe it, although perhaps it's the fault of his translators. Can a more informed person explain his archaeological evidence for such a movement of people? Does he propose a movement directly north through the Caucasus, or is it around the Caspian on the east and then onto the steppe?
Whatever the precise route, I could see it for the "precursor" language, but given the ties to Uralic, I don't think the Indo-European languages themselves could have spread in this fashion, particularly not if Indo-Iranian is held to have peeled off first. Also, the Gramkelidze Ivanov model suffers from the fact that a movement of "Anatolian" to the Balkans, leaving the rest of the language speakers to develop the language in eastern Anatolia before their counter-clockwise movement around the Caspian and then a movement of Anatolian
back to Anatolia is, as Mallory points out, way too convoluted without any evidence to support it.
Also, in order to go this route, wouldn't it have to be the case that people living 'cheek by jowl', i.e. in very close proximity to one another in eastern Anatolia would have to have been speaking very different languages, given that there were Urrartians, Hurrians in the area speaking very different languages, languages that are neither Semitic nor Indo-European. Unless, perhaps, these languages were later arrivals?
https://en.wikipedia.org/wiki/Urartian_language
Anyway, this has nothing to do with the "Armenian" language, which Anthony and a lot of linguists believe entered Anatolia later and through the Balkans. Well, Ivanov et al leave open the possibility it, and Greek made a run along the north shore of the Black Sea, yes? This where some more ancient genomes will help as well.