Except the directional flow of what you're describing here is exactly opposite of what the archeological evidence shows.. with clear material cultural continuity in an eastern/south-eastern direction from the Black Sea to south Asia.

It's likely that when the Indo-Iranian languages arrived in south Asia they were already in the process of heavy fusing with local cultures, esp in the BMAC. References in Sanskrit writings about events that may precede Sanskrit speaking doesn't prove anything. There are plenty of cases of elite language replacement, and it's likely an elite class spoke IE and potentially even translated and wrote down non-IE oral histories.

Cultures are not necessarily languages or "peoples." Language replacement is a common phenomenon. India can be India with an insanely long deep amazing history that the world recognizes as rich and powerful and ancient... without being the origin of one of the languages spoken there.

In any case, non-IE languages continue to prosper in the subcontinent, and probably did even more so among regular people back then. Which is very much not the case for the central European region, unless you count the crazy linguistic patchwork in the Caucasus (which is heavily divided up by mountains).

Anatolian divergence happened very early, so much so that it has an entirely different gender system than all other IE languages. And lacked common IE words related to horse riding, chariots, metal-working, etc. By the time of the written documents you describe, it had already diverged into several distinct languages, implying a long history in the region. Likely a founding population that made its way around the Black Sea on some path.

And a similar phenomenon to India: established itself as a local elite over a population that spoke a different language, and kept oral histories and eventually wrote in their own language about the gods and stories of the people they fused with.

Anyways, the strongest evidence is in the common IE vocabulary -- esp around plants and animals -- which is strongly biased towards the temperate climate around the steppe.