Splitting the argument into two pieces is most concise:

1) Sarasvati river paleochannel radiocarbon dating of Rig Vedic Sanskrit (<=3000 BCE)

I talked about this here: https://news.ycombinator.com/item?id=42330343
This is by far the strongest piece of evidence that the Kurgan model for IE language diffusion, which has Indo-European languages entering India circa ~1500 BCE, is grossly incorrect.

We now know the Sarasvati, the most prominent river mentioned in archaic Rig Vedic Sanskrit texts, began to desiccate and disappear in 2600 BCE. That desiccation process is actually mentioned in later Sanskrit works like the Mahabharata, in Sanskrit that has significantly linguistically evolved.

There are hundreds of urban sites along the now-dried Sarasvati riverbed which have now been discovered, like Bhirrana, with cultural continuity back to 7500 BCE. But let's steelman the Kurgan model and assume there was a replacement of this advanced urban society's language by IE-speaking Steppe nomads, that left no archeological trace whatsoever. This still means that Indo-European languages were in India at least by 3000 BCE, making this the oldest attested Indo-European language in the world.

I would challenge you to explain how the Sarasvati paleochannel evidence doesn't completely break the Kurgan model. Even leaving aside the Sanskrit corpus which does not remember any migration or original homeland of the Aryans before India.

2) Anatolian languages (1800 BCE)
The earliest hard chronology of Hittite is the Anitta text of the Kussara (1800 BCE). The linguistic analysis (such as it is, linguistics isn't the hardest science, much less so than radiocarbon dating) implies this IE branch diverged early. Like the dating of Rig Vedic Sanskrit to at least 3000 BCE, it's likely the Hittite language was evolving for hundreds of years before this.

Did IVC settlers directly migrate to Anatolia? Was there a cultural domino effect spilling out from the IVC to the BMAC, then either north through the Caucasus or south through Iran / Mesopotamia? We don't know.

We do know there were widespread cultural and economic ties between India and the Middle East at least as far back as 3300 BCE. IVC seals and finely-wrought carnelian beads were discovered in the royal Sumerian tombs in Ur, and many texts described the thriving economic trade between these two regions. Indian DNA has been found in Syria in 2500 BCE. The Mittani in ~1500 BCE are an even clearer example, with Rig Vedic deities, a military elite, and a horse-training text that are distinctly Indo-Aryan, not even Indo-Iranian.

In other words, cultural and human diffusion from India to the Middle East was already happening hundreds of years before the first attested Anatolian IE language is found in the Hittite.

And we know it was happening in the other direction (east) as well. The Tarim Basim mummies had Indian genetic markers, implying the Tocharian branch of IE may have also arisen out of India.

The IVC culture had a bigger population than Egypt and Mesopotamia combined. It was an advanced and urbanized culture. There were major environmental shocks that caused mass migrations. There has been migration out of India from at least 3300 BCE to the common era with the Gypies / Romani.

Is it so hard to believe IE languages could have come from such a place?