That is about 2500 years after the period we're discussing, and in a region conventionally considered to be on a different continent. It isn't a mere presumption that the Kurgan culture didn't have writing; archaeologists have been looking for it diligently for more than a century and have found extensive collections of well-preserved grave goods, but no writing. Writing was invented about 1000 years later in Sumeria, probably in Egypt, and possibly in South America, but not in the Lower Volga homeland of the Proto-Indo-Europeans. (The North American and Chinese inventions of writing seem to have been independent, but were another 2000 years later still.)

The Hittites adopted the Sumerian form of writing; they did not bring a writing system with them from the Volga. Neither did other Indo-European groups have writing, which is why Hittite is, as you say, the oldest attested Indo-European language.

The Hittite documents, besides recording several Indo-European languages from the same (Anatolian) branch of the Indo-European language family, also record some fragments from an Indic language, making that the older attestation from another Indo-European branch than that of the Hittites. (The next attested Indo-European branch is Mycenaean Greek).

That Indic language was the language of some group of people who at some point in time, perhaps after a war victory, had become the main members of the elites who ruled Mitanni, a Southern neighbor of the Hittites, located mostly in present Syria, where most inhabitants were speaking Hurrian, a non-Indo-European language.

Those Indic-speaking people were renowned as expert horse trainers, so the quotes from their language were encountered in Hittite documents about horse training.

Most known data is consistent with an older migration towards South Asia of the people speaking Indic languages, who had gone both towards East, reaching India, and towards West, reaching as far as Syria, where they entered in contact with the Hittites and other related populations, who had migrated towards South at an even earlier date and through a different path, reaching present Turkey.

The Indic migration has been followed much later by a migration on the same path of people speaking the closely related Iranian languages, who have reached the present territories of Iran, Afghanistan, Tadjikistan, forming the ancient Persian empires, after various conquests.

The people whom we now name Hittites used another name for themselves, and they called Hittites a non-Indo-European population, who were the former inhabitants of the territory ruled by what we call Hittites.

There is some evidence of at least proto-writing existing in the “Old” European societies that the Indo-Europeans replaced prior to 3500 BC. Of course no indication that it was preserved or further developed.

https://en.m.wikipedia.org/wiki/Vin%C4%8Da_symbols

It should be noted that there is a very great difference between "proto-writing" and writing.

It is likely that various kinds of "proto-writing" have been independently invented in a lot of places, but very few of them have evolved into writing systems.

"Proto-writing" is just a set of graphic symbols that are used to designate various things. Such a set of symbols can be used e.g. to write an inventory, to tag things to show ownership or purpose, to show on a map what can be found in certain places, and so on.

"Proto-writing" cannot be used to write human speech. All systems of "proto-writing" that have evolved into writing systems have done that by reinterpreting a part of the graphic symbols, or sometimes even all of them, to no longer be the names of some things, but to have a phonetic meaning, i.e. to represent some sounds of human speech (syllables in almost all cases), allowing thus the writing of the more abstract components of the speech, like various grammatical markers.

Therefore for a system of "proto-writing", it does not make sense to ask which is the language that has been written with it, because there exists no such language.

The only kind of information that can be known about a system of proto-writing is which is the thing denoted by each symbol. Even when the meanings of all symbols are known, that does not offer any information about the language used by those who have invented and used that system of proto-writing.

For now, there is no evidence that the Indus script was a writing system, because only very short strings of symbols have been preserved. It could have been a writing system, because by that time other writing systems already existed not far away, which could have inspired them, or it could have been just a proto-writing system, which would give no clue about the language of its users.

Yeah, I was thinking about mentioning the Indus script here. Lately I've been thinking about brand logotypes, hallmarks, chops, and cattle brands as potential exemplars of proto-writing (though of course in our culture they are heavily influenced by writing); maybe the Indus valley script was used to mark pots as being made by a particular potter, for example? Or dedicate a temple to a particular god?

The Proto-Indo-European language is usually dated to something like 6000 years ago, well before any writing.

So? What about the Hittites? There is a slight gap between 1700 BC and 4500 BC.

Prior to the discovery of the Hittite language, linguists had compared the various Indo-European languages they knew of and did much of the work of reconstructing the Proto-Indo-European language based on comparative linguistics. This work was highly conjectural, but it provided something akin to a falsifiable theory that could be tested by the discovery of another written Indo-European language. Such a language was Hittite, and the Hittite language fits the model of Indo-European languages that had been constructed prior to its discovery.