Why assume that "social progress would happen" without rational planning, as though the natural state of affairs weren't the stagnant dominance hierarchy so beloved by conservatives? If that's the automatic flow of human history, it would had to have been sustained by a supernatural force like God (as Burke would have said), which would be absurd.
You say this progress should happen slowly rather than all at once in a revolution, to benefit "the whole of society." But in a conservative society (a monarchy, plutocracy, dictatorship, etc), most people are poor and powerless. They're precisely the folks who would prefer immediate social improvements.
So what you mean to say is that progressive revolutions are bad specifically for the power elites, for the minority at the top who exploit the poverty and powerlessness of the majority. And that goes without saying, but so what? Burke will pretend he's fighting for stability for the benefit of the whole society, whereas he's really fighting for his dominance. He thinks the aristocrats know better than the peasants about what's good for the latter.
So his conservatism is condescending and elitist. I'm not opposed to elitism in some areas, but the problem is that political elitism is a mixed bag. The aristocrats obviously have better education than the illiterate peasants, but they're also more corrupted by their power so it's not obvious their opposition to revolutions is based on elite understanding or on ulterior motives.
You point out that the French Revolution involved mass murder. Yes, I'm sure that horrified Burke. But why wasn't Burke horrified by the mass murder required to keep monarchies stable? Think of the wars of conquest, the torture and imprisonment of free-thinkers and dissidents, and the oppression of women and the poor? As I say in the article, it's an ocean of blood compared to a pin prick. So forgive me if I doubt Burke's opposition to progressive revolutions is entirely a moral or humanitarian one.