It's worthwhile to outline the basic tenets of the Aryan Invasion Theory (AIT) at the onset. As you mentioned, it supposes that the decline and fall of the Indus Valley Civilization (IVC) is directly linked to the invasion of Indo-Aryan speakers on horseback, who waged war and overthrew the IVC cities. Proponents of this theory pointed to the mention of what seem like enemy "cities" ("pura") mentioned in the Vedic texts, and hostile Non-Aryan tribes that could not be reliably identified. Within the corpus of Vedic literature, the Indologists read the struggles between the noble "Devas" & demonic "Asuras" as an allegorical rendering of the wars the early Aryans must have encountered upon their invasion of India. The natives were subjugated and subsumed within the hierarchical and endogamous caste system created to ensure strict separation.

It's also worth noting the alternative to the AIT that is being propagated by the people mentioned in the video you've linked: the Out of India Theory (OIT) which states that Indo-European languages were seeded by migrations out of India. In this retelling of history, the IVC was Vedic in nature, and the people spoke Sanskrit. This theory supposes that Sanskrit (or proto-Sanskrit) was native to India, and has roots in India back to the Paleolithic Era. The success of the IVC led to its people, culture and language being carried across Eurasia, surviving today as the widespread Indo-European languages. Proponents of this theory point to possible similarities between the IVC & Vedic culture, such as the extensive "Baths" in IVC cities, the "Pashupati" seal, and the "Priest" bust. The OIT has great traction among many Indians, including leading academics.

Now, let's consider the facts as we can establish them, and how they impact the tenets of each theory. For one, there is a considerable gap in time between the decline of the IVC and the observable evidence for Indo-Aryan cultures. The IVC cities appear to have entered into a deep decline starting around 1800 BCE. The IVC were a Bronze Age civilization and didn't have horses (domesticated in Central Asia), significant iron usage, or the chariot (the IVC had bullock carts). Evidence for these and other distinct cultural goods associated with Indo-Aryans show up in the archaeological record of North India & Pakistan only around 1300 BCE. The AIT notion of the Indo-Aryans overthrowing IVC cities like Harappa doesn't work because Harappa had already fallen centuries before. Similarly, the OIT relies heavily on the IVC being a Vedic Civilization, something that remains unsupported by any hard evidence. Nor does it offer a credible alternative model for how Indo-European languages could have been seeded by migrations from India. Such large scale migrations are neither reported in Vedic literature, archaeological evidence along the supposed routes nor noted by other contemporaneous populations. The OIT also accords Sanskrit a much more fundamental basal position in the Indo-European (IE) language family than most linguists do.

Let me offer a more widely accepted softer alternate theory, the Aryan Migration Theory (AMT). Informed by recent population genetic evidence (2019), the model is broadly thus:

10,000 years ago, South Asia was populated by hunter-gathers, distantly related to the Andaman Islanders. Referred to as the Ancient Ancestral South Indians (AASI) in the genetic literature, these people had deep roots in India. Around 8000 BCE, an Eastern branch of Iranian Farmers arrived in the Indus Valley. They carried with them the technological toolkit for wheat & barley, which had been domesticated from their wild Fertile Crescent ancestors. The Fertile Crescent of Iraq & Iran receives most of its annual rainfall in the winter and as a consequence, the crops of the Fertile Crescent toolkit were heavily dependent on winter rainfall. The Indus Valley marked the transition into the different climate zone of South Asia, where most rainfall occurs in the summer monsoon season. The Iranian Farmers could penetrate no further into India, and settled along the Indus & its many tributaries. Here they encountered the native AASI hunter-gatherers and intermingled with them. It is this hybrid population that gave rise to the Indus Valley Civilization. We have genetic evidence from skeletons that suggest the IVC people had somewhere between 10-50% AASI ancestry.

At some point around 1800 BCE, the IVC enters a deep decline. The reasons are poorly understood, though the popular current theory points to rivers drying as a major cause. When the IVC was established, the Yamuna used to flow east and join the Indus. Around 4000 years ago, tectonic shifts raised the river bed and diverted the Yamuna westwards, joining the Ganga. Similar changes affected the Ghaggar-Hakra, which dried up almost completely. The now mostly dry channel of the Ghaggar-Hakra had supported multiple IVC sites. Whatever the cause, the IVC was long gone by 1500 BCE. Around this time, the Indus farmers had managed to developed wheat & barley varietals that could tolerate summer rainfall and they moved into the Gangetic plain. Within a brief period of time, their population exploded, spreading into mainland India. As these farmers moved into new territories, they encountered the AASI hunter-gatherers of the region, incorporating them over time. The Dravidian languages probably appear sometime after this.

At roughly the same time, the Sintashta people of the Central Asian steppes were migrating outwards from their homeland. They had acquired horses & chariots from other Steppe peoples and migrated southwards. At some point near modern-day Tajikistan, they split into two groups: one moved down into modern Iran and gave rise to the Iranic people, and the other branch moved into the Indus Valley as the Indo-Aryans. As these pastoralist nomads moved, they encountered farming communities, incorporating them. We have skeletons from the Swat Valley of Pakistan from around 1100 BCE which show a largely Steppe population with some Iranian farmer ancestry. As these waves of migrants moved further into India, they met and merged with existing populations of Iranian farmer + AASI. These Steppe migrants spoke an ancestor of Sanskrit and practised an early form of the Vedic religion. The Vedas likely achieved their final form in the Punjab and their Vedic culture would become dominant across the Indus & Gangetic plains. Modern day South Indians are predominantly AASI (up to 50%)+ Iranian Farmer (10-40%) + Steppe (up to 10%), while Modern North Indians are predominantly Iranian Farmer (up to 50%) + AASI (up to 30%) + Steppe (up to 30%). The ratios of each depend on region & caste to some extent.

The "Aryan Invasion Theory" is not a formal theory and as such is not well defined enough to be either proven or refuted. In the popular imagination it is the concept that a race of fair-skinned people migrated into the Indian subcontinent in pre-historic times (since we don't have written history from then) and displaced the native inhabitants, who were pushed into Southern India. Of course this is not true as the construct of race itself is artificial and has long fallen out of use by serious historians.

This idea first came into popular existence in the 19th century. In the late 18th century, Indologist William Jones showed similarities between Sanskrit, Latin and Greek, and suggested that all three had a common origin, a single ancestral Indo-European language. It was not a terribly original idea that various European languages and some others like Persian, had a common origin, possibly originally spoken in the Caucasus Mountains. That Sanskrit ("the Samskroutam language") was one of these languages had been previously suggested by Pere Coeurdoux in 1768. But Jones was the first to show this with some evidence, most memorably during this landmark address to the Royal Asiatic Society of Bengal in 1786.

Now why did this original language come from the Caucasus mountains? I am not sure, but even up to Jones' time, many European scholars tried to contort their theories to match the Biblical narrative that all humans originated from Mount Ararat (historically in Armenia, today in Turkey) where Noah's ark ended up after the flood. So it might have been a hangover of that thinking. It's also a geographically convenient location for an outward migration into Europe and the Near East.

But there was also a strong counter-movement to the Biblical theme at that time, and some of these scholars took a very different line upon learning of the linguistic connections to Indian languages. The French astronomer, Jean Sylvain Bailly, in 1777 (even before Jones' announcement) placed the earliest humans on the banks of the Ganges, and stated that, "The Brahmans are the teachers of Pythagoras, the instructors of Greece and through her of the whole of Europe". Voltaire agreed with him. Proponents of India being the cradle of civilization would find many famous voices, all the way to at least 1855, when Lord Curzon, the governor-general of India and eventual chancellor of Oxford would say that, "the race of India branched out and multiplied into that of the great Indo-European family". This "race" had spread by conquest and "obliterated nearly all traces of the former non-Aryan languages".

The view was not without some (apparent) foundation. While Jones believed that Sanskrit, Greek and Latin, descended from a common language, other linguists felt that Sanskrit was most like the original, with some (like the early Theosophists) insisting that, "Old Sanskrit is the origin... of all modern European tongues and dialects''. Jones' original view would come to become the dominant one by the late 19th century thanks in part to the work of linguist F. Bopp.

The India-origin view was full of holes from the beginning, as some articulate scholars pointed out. It was also hated by those despising kinship with "inferior races", and evangelists who found it inconvenient for their messaging. Moreover by the mid-19th century, the British Raj was firmly established in India, and it was more convenient to start promoting Indian history as a series of invasions and foreign rule.

So by the late 19th century the dominant view was best articulated by F. Max Muller, who is one of the most prominent proponents of what would come to be called the "Aryan Invasion Theory":

1. The Aryans had originated in Central Asia, with one branch migrating to Europe and another settling in Iran. A segment of the Iranian branch subsequently moved to India.
2. The Aryans invaded in large numbers and subordinated the indigenous population of northern India in the second millennium BC
3. The invaders imposed their language, composed the Rig Veda, brought in a superior civilization, and imposed the caste system to maintain racial purity of the upper castes (the "Aryans")

This was a very race-based view of language-based evidence - a fallacy which Max Muller did try to correct later. But the original view was too popular by then.

The evidence pointed to a people speaking a (hypothetical) language called Indo-European or Indo-Aryan, following roughly the movement in point 1: a gradual migration of a nomadic people into the Indus region via present-day Afghanistan. Rajesh Kochar in "The Vedic People: Their History and Geography" says that the Rigveda and the Zoroastrian (Persian) holy book the Avesta, have references to the geography of Helmand province in Afghanistan, and to a similar nomadic culture. The later Vedas in contrast appear to be an amalgam with local cultures in North Western India (probably the remnants of the Harappan civilization). So that debunks the narrative of the conquerors obliterating the local languages and culture. The finding that most languages in northern India (the subcontinent) belong to the Indo-Aryan family, while those in the South, like Tamil, do not, was taken as additional proof for the invader-displacement theory. This does not mean there are racial differences between these regions, and moreover languages spoken by people always change over a period of time. There was never evidence for a large-scale invasion that caused massive population/language/cultural displacement.

Though today this theory is seen as a colonial attempt to demean Indian civilization, at the time it was those with a racial superiority complex that were most upset with Muller. In India, on the other hand, the idea won a lot of support. E.g. S.C. Banerjee claims in "The Development of Aryan Invasion Theory in India", that the Brahmo Samaj, a prominent socio-religious reform association, played a major role in promoting it. Jyotibha Phule, an authority for Dailts, used it to claim the Brahmins were the Aryan invaders, and thus alien to India.

A couple of additional references:

• The Quest for the Origins of Vedic Culture: Edwin Bryant
• The Penguin History of Early India: Romila Thapar