we don't actually have the original documents, and those we do have have gone through countless revisions

This actually isn't thought to be true of the Qur'an, if by revisions you mean later insertions and changes made after Muhammad's death.

That idea, that the Qur'an was revised or compiled over time was once put forward by revisionist textual critics of early Islam, most notably John Wansborough in the 1970s. Some of his students like Patricia Crone and Michael Cooke followed his ideas.

But the discovery of very early Qur'an manuscripts, like the Sanaa manuscript, which are sufficiently close to the received text that the errors contained can usually be attributed to copying errors or differences in orthography, has made that position basically untenable. Not to mention that Islamic tradition itself maintains that there some Qur'an variants (i.e. of the mushaf) prior to the creation of the standard Uthmanic codex, which didn't so much "revise" as, at least according to tradition, standardize the variants.

As such the general consensus is that the Qur'an is a primary source. That is, it is what it claims to be: a text produced by the Prophet Muhammad.

The problem is, what kind of primary source is it? Readers more familiar with the bible are used to a holy book that is basically a chronological story. It starts with Creation, Moses shows up with some tablets, Israel kicks around for a while, they all end up in Babylon, some prophets say some things, and then boom Jesus shows up in the sequels, he dies, he comes back, and so on.

The Qur'an isn't like that in either content or structure. It's arranged not chronologically, but by length. The revisions, that we do know about as made by Muhammad himself mean revelations in particular suras might have chronologically come later than the text around it. There are very few proper names of either people or places and almost no stories, although there are references to stories. There are only a couple references to contemporary events and these are oblique. Many parts are esoteric or if not entirely obscure, as in the case of the mysterious letters.

The texts designed to explain these esoteric portions, the exegesis, were written down 100-200 years after the fact, and so lose that primary source quality entirely.

So is the Qur'an a primary source? Sure, but it's not necessarily an especially useful one for the history of its time. That of course doesn't make the text any less interesting for historical study, especially given just how many unanswered questions we have about it.

For further reading, I quite like Neal Robinson's Discovering the Qur'an: A Contemporary Approach to a Veiled Text as an introduction to what the Qur'an is and how it's structured.

For some historical discussion see, for example, The Qur'an in its Historical Context edited by Gabriel S. Reynolds.