I have a general idea that there was a considerable amount of conflict in the late antique Roman empire over whether Christ had one nature, or separate divine and human natures, and this conflict was widespread throughout society.
To me, this seems like a pretty arcane theological point. Why was it such a big deal? Why did anyone who wasn't a theologian care? Or was it limited to relatively few elites?
My only good point of comparison is the disputes that led to the Protestant Reformation, but there there are obvious reasons for conflict, in that Luther was challenging the authority of the Church.
It may be that I'm misinformed on the historical background, in which case I'd appreciate being corrected!
It's a multifaceted set of interrelated conflicts.
Firstly, theologians considered that it mattered for two main reasons. 1) getting God 'right' mattered because if you were wrong in your doctrines about God you were worshipping incorrectly and this was tantamount to idolatry. 2) getting theology about Christ 'right' mattered because if you got it wrong your understanding of salvation, how it worked and what it was, depended on it. And consequently, getting this wrong basically destroyed your system of salvation.
So there was a lot at stake doctrinally. These were not arcane abstracts, but fundamental and basic features of Christian theology that had to get nailed down.
Secondly, the conflicts often had political dimensions related to geography, church centres, bishops, imperial favour, and historical traditions. These were all overlaid with the theological concerns, and ultimately inseparable. It matters, for example, if a council meets and declares your whole region to be out of communion because they decided your bishop was a heretic.
On the one hand, it was primarily a discussion among theologians and elites, but it wasn't entirely confined to them. However, documentary evidence for 'ordinary Christians' is much more limited, because (perhaps obviously) people were most interested in preserving theological treatises. So our evidence is skewed.
To go back to your initial questions. If Christ has only one nature, is he God, or is he human? If he is God, what's the incarnation all about? If he's human, how is salvation possible? And isn't worshipping him idolatry? If he's a hybrid with a single nature, then he's not God and not human. So again, not worthy of worship, and not able to save.
These are the kinds of issues at stake.
Is it such an arcane point? It is still a source of great debate between people today.