I think that's an interesting idea, but I'm not so sure about it. I do think Armenian, and possibly Greek too, were in contact with the Hittites, as well as the Hurrians (not only the Urartians, but Hurrians-proper), and quite likely Hattians.
I think the Hattian gods may have been Indo-European, or they contributed to an early, unsplit IE culture (maybe Greco-Armeno-Aryan or an Anatolian-Greco-Armeno-Aryan group). Arinna clearly has the IE "ar-" sun prefix, which is present in Armenian, Anatolian, and Indo-Iranian languages. Estan (Hittite: Istanu), the sun god, reminds me of Armenian Astuas. Taru is of course related to bulls (this connection is well established). There are a few others ones I suspect as being Indo-European as well. We could suggest that these were loans from Hattian>Anatolian>Armenian but it doesn't explain how the Indo-Iranians got these roots unless the Indo-Iranian homeland is in Asia Minor and not Central Asia. All of these are thought to be PIE roots, which would suggest that, either the PIEs were in contact with the Hattians, or later IE peoples introduced these to the Hattians. Either way, it means that IEs were in Hatti by the end of the third millennium BCE.
Hurrian Ulikummi is probably from Armenian Uelik, which would be Geghik in modern Armenian. The Hittite version is Illuyanka. I'd think the Hurrian was from Anatolian, but the -ik is an Armenian suffix (ummi is a Hurrian suffix, but the meaning is unknown, as far as I know).
What are your thoughts on all this? Can we create a potential model to explain this?