My objection is based on geographic grounds. My saying "Out-of-India," you give the impression that the entirety of the Indian subcontinent was the Aryan homeland, when in fact even the so-called OIT posits only Sapta-Sindhu as the Aryan homeland (since that is where the early Vedas are situated). This means that you still need to account for how Indo-Aryan cultural and linguistic expansion occurred in the 90% of India that lies outside of Sapta-Sindhu.
No one claimed that English would be closer to Sanskrit than Kannada (or any other heavily Sanskritized Dravidian language). The point is that even after thousands of years of Sanskritic influence on Kannada and other Dravidian languages, the difference between Kannada and Sanskrit is still far greater than the difference between Sanskrit and ancient Iranian languages. What this demonstrates is that mere cultural and religious influence cannot account for the Indo-Aryan expansion throughout Eurasia, because even if Vedic Hindus had some great cultural and religious influence throughout Eurasia (for which there is absolutely no evidence), the most that we could expect from such influence is a Sanskritized language like Kannada. It cannot explain the wholesale replacement of pre-existing languages and the emergence of Iranic and other Indo-European languages throughout western and central Asia.
a. They belonged to the same sphere of Vedic civilization, although far from the center, leaving enough scope for change in dialects.
b. The words that they share belong to a superset of words from Vedic that also has words shared by other languages of India.
Do you think the European languages share the same closeness with Vedic as much as ancient Iranian languages do? If not, why? How are they different in being different from Vedic as compared to South Indian languages?
What languages are you talking about? I am not aware of any Dravidian languages existing "in the heartland of Indo-Aryan languages." Dravidian languages in the northern half of the subcontinent only exist in peripheral areas, such as the Brahui in Balochistan (western Pakistan), and they are heavily influenced by their neighbors. It's also likely that they are recent migrants, rather than the remnants of some ancient Dravidian population.
You are simply shifting the goalposts.
1. If the split between the Vedic and Southern languages happened "further back in time," how do you propose that initial expansion took place? How was it qualitatively different from the later Aryan expansion?
2. If the Aryans had already expanded throughout India in the past, why didn't they demonstrate any basic awareness of the larger Indian subcontinent in the early Vedas? In the Classical Hindu epics like Mahabharata and Ramayana, there is a clear awareness of the Indian subcontinent as a wider entity. Where is the awareness in the early Vedas? Why don't we have mentions of regions like Bengal, Kalinga, or Andhra in the Rig Veda, if the people who wrote Rig Veda had already spread their languages throughout India?
Europe is one of the most fertile continents in the world, and Mediterranean Europe in particular (which received the heaviest migrant flow from Fertile Crescent) has hosted dense agrarian populations since antiquity. In contrast, central Asia and eastern Iran have never been densely-populated by agrarian populations at any point in history, as environmental conditions have not permitted the creation of an enduring agrarian civilization in this area. The peoples who successfully spread their languages throughout the central Asian region, such as the Turkic peoples, could do so because they were highly warlike, nomadic, horse-riding peoples, and they were not looking for farmlands.
1. Getting ousted in power struggle
2. Population explosion
3. More fertile areas
4. Geographical events
5. Trade contacts
If Fertile Crescent had developed agriculture (and actually the one that introduced agriculture to Europe), and the destination being fertile than the source is the only criteria, there was no need to move. However, as I mentioned above, people move for variety of reasons.
The case of Mitanni is proof against OSST, not proof in favor of it, because the Mitanni expansion was not accompanied by any linguistic shift whatsoever. In fact, it was the Mitanni themselves who adopted the local Hurrian language. Thus, wherever the Mitanni came from and whatever the Mitanni were doing, it was insufficient to explain the kind of transformation that we see throughout Eurasia by Indo-Europeans. The nature of the Indo-European expansion must necessarily be far above and beyond the Mitanni expansion. Moreover, unlike in the Middle East, the steppes of central Asia already had horses and chariots for centuries. Do you propose that it was Indo-Aryans who introduced horses and chariots to the peoples of central Asia?
If Indo-Aryans had dominated all of central Eurasia and spread their languages throughout this region, we should certainly expect to see religious traditions directly derived from Vedic religions in this area (as opposed to simply sharing a common ancestor with Vedic religion). Where is the evidence of such religious traditions?