The Mitanni are a unique entity that emerged in a very different time period. Its framing was different than any other realm in the ancient Mid East. It acted as a royal caste and noble warrior elite controlling an otherwise patchwork of some 300-500 kings and city states. The connection between all was a military that coordinated coalitions and a series of Hurro-Aryan cultural conceptions centered around this central royal and noble warrior caste.
The Medes by contrast were an Assyrian rebel state that had its own political systems, likely influenced by Elam. Persia was essentially and IE Elam with traits unique to their region, but it is the closest thing we get to Elamite without being Elamite.
The Scythians may be able to do this, they could be much like the Kassites or Arsacids. However, where will they rule? They would need to make some great gains in order to acquire all three areas. This means destroying the Medes and the Persians in order to protect Urartu and Elam and then ideally driving the Medes forth and then subjugating the more sedentary Persians. This is a possibility. The Arsacids managed to place Akkadian on life support and once their benign rule was removed by the Sassanids, Akkadian evaporated in less than two centuries.
But all of this is to say, each of these areas are under great threats and are already inundated. Hurrian as we mention is mostly gone from Syria by the year 630 BCE, with its only holdouts probably existing in the city of Carchemish. Perhaps with recent and hoped discoveries of Washukanni, we can decipher how long Hurrian remained in the region as an Assyrian province. My estimation is that it was destroyed during the Bronze Age Collapse and very early Iron Age, 1070-950 BCE. Hurrian as spoken in Urartu was probably a deep minority even in 750 BCE at its height. The eclectic nature of Urartian monarchical display tells us the story that their realm was interested primarily in prestige displays when it came to language, a custom common in their area. Luwian, Akkadian Ardinian, and so forth were all used in their realm, despite the first two having essentially no speakers in their home range. Considering the formation of the kingdom as an alliance between many ethnic groups and tribes, it would make sense that they choose frankly foreign or semi-foreign prestige tongues for their administration.
Elamite had been in dire straits ever since 1114-1110 BCE when Anshan was destroyed by the two Akkadian monarchs, Nebuchadnezzar I and Tiglath-Pileser I. It was in that frame that we see the appearance of presumably IE elements in the east, first as a minority and then over time overtaking the Elamite population. Meanwhile in Elam, the kingdom was becoming an Aramo-Elamite kingdom, with Aramaen-Chaldaean-Piqudu tribes ruling the government and having great sway. Aramaens had migrated into Elam and were the majority population along the borders of Elam withe Mesopotamia. In the north, the old Elamite vassals made up of a diverse mountain folk people providing it goods and soldiers were conquered by the emergent Median states which were in turn countered by Assyria and Urartu, with Elam playing no role in stopping them. Elamite despite its great fame and prowess in the past, was not even used by the Arsacids or the Greco-Elamite kings, who used Aramaic for administration. Great changes had occurred essentially, Elamite lost its prestige as early as 400 BCE most likely, only 130 years since its inundation under the Achaemenids. It would seem thus, it was already becoming extinct in its homeland due to low rates of birth, destruction in war and migration waves that in turn assimilated the Elamites.
Mesopotamia is a known quantity. Without Assyria enforcing an Akkadianism upon the region, it was an extremely tense ethnic situation. Aramaens and Akkadians battled viciously for authority over Babylon with signs of ethnic strife until the Assyrians asserted itself and placed only Akkadian kings on the throne. In fact, all rebellions against Assyria in Karduniash, were the result of this Akkadian-Aramaen battle over influence. One that the Assyrians were also embroiled in, as the Aramaen tide had overtaken many areas of traditionally Akkadian or Hurrian territories (the two languages that we may say are Assyria's primary cultures). When Assur-dan II (934-912 BCE) ascended to the throne, his annals make claims that the western enemies or the Aramaens had massacred Akkadians in the west and enslaved thousands of his people and that his military campaigns westward were intended to free them from slavery and restore an Akkadianism unto the region, framing the wars in a light of ethnic strife.
The Scythians will have as much difficulty maintaining all three lingustic spheres. I think they can maintain Akkadian, and possibly Hurrian, if they can act fast enough and destroy the Medes. Regarding Elam, I feel it is a lost cause for them to be saved linguistically. They have too much incoming migrants and their language holds prestige among people who do not even speak it, that is it is already a ceremonial language in Fars. In Elam proper, it is in a very small area and one that is already covered in migrants and their self-sustaining parallel societies. Indeed, the Iron Age and Bronze Age Mid East is not a good example for cultural and ethnic diversity in a harmonious sense.