There hasn't been much historiographical debate over the centrality of slavery since the middle of the twentieth century. Previous to a few decades there, which overlap heavily with and is related to the height of post-Reconstruction lynching, it's all slavery still. But even in that, the arguments clearly revolve around slavery. The question is more of how slavery caused the Civil War than if it did.

Very briefly, and neglecting the black radical tradition because I know significantly less about it, the historiography works out like this: For the rump of the nineteenth century you have actual participants doing the history. These are Lost Causers on the Confederate side and triumphalist antislavery people on the other. The Lost Causers lie flagrantly about their motives in order to rehabilitate their image. You sometimes see cracks in the histories about how they're better witnesses for the opposition than themselves.

Around the turn of the century, the nationalist school arises. The nationalists insist that slavery is at the root of things and the South was broadly in the wrong during the war, but totally right on Reconstruction because everyone knows black people are just inferior. So now we can all get along. The historiographies of Reconstruction and the Civil War tend to run in similar lines, for the obvious reasons. If you know your Reconstruction historians, this is also when we have the Dunning School.

The nationalists see challenges from the progressives, who are big proponents of economic interpretations of history and frequently indifferent to hostile to the examination of ideas which aren't immediately related to economics. They argue that the Civil War was caused by, variously, the tariff and competing philosophies of development. There's a tendency to just stop there, rather than acknowledge that to the degree these divisions are supported by the texts (which is dubious) they are also profoundly slavery-inflected. The philosophy of economic development in the South literally is slavery: the more the better. Likewise antislavery whites argue that slavery is a drain on the economy and free labor is inherently more efficient. So the economic argument is inescapably a slavery argument.

After WWI, as part of the general hostility to war and disillusionment we have the growth of the needless war/blundering generation school. They generally believe that slavery would have soon died a natural death anyway and had reached its physical limits, so all agitation against it was just the work of cynical pols irresponsibly scrounging up votes and thus driving the nation into a tragic, pointless war. The war could have and should have been averted. Some of them will go on to say that since slavery was destined to die, everyone knew that and therefore something else (anything else) had to be at the heart of things.

After WWII, things shift around again. Now you see the irrepressible conflict school. They believe that there's a fundamental conflict between slavery and free labor which might be contained or delayed, but would have come to blows eventually. You could not put a cork in it forever. They go around with the needless war types for a few decades, overlapping with the next group.

This is about the point where I begin to look at general histories of the Civil War as potentially useful texts today rather than a kind of primary source in themselves. For narrower or neglected subjects I'll go back further of necessity or out of historiographical curiosity, but I tread with much more caution than I might otherwise.

Roughly contemporaneous with the Civil Rights Movement you see a school of neo-abolitionists. They rehabilitate the antislavery movement from decades of abuse by basically everyone, recasting the Civil War as not just a war that had to be fought, but also one that was good and righteous to fight. That's compatible with the irrepressible guys (most of these things are separate, if interrelated debates that I'm breaking up chronologically for clarity) but significantly to one side. The neo-abolitionists exist in tension with the consensus school otherwise dominant about the same time which insists that US political differences are generally much less significant than a truly broad consensus.

By this point, everyone is pretty firmly on the same page with slavery. The needless war/revisionist types could talk around it sometimes, but I think that taking them seriously required a very particular alignment of black marginalization and attendant racism that just isn't credible anymore. There are attempts to recast slavery in different terms as cultural conflicts or argue that it's not necessarily a big cleavage to ordinary white people, but still in the HowSlaveryDunnit tradition rather than the If.

For a sampling of all this, except the most recent stuff, you can check out Stampp's The Causes of the Civil War. I have some gripes with his editing and organization, but it's essentially a collection of documents by historians and historical figures on the question.