Mmm, I have been avoiding the centaur battle. But this seems like rules here. Difficult terrain can be both: movement reduction and a skill challenge.
The point is that "difficult terrain" (in those words) is a defined game term, and has been the same in 3e, 4e, and 5e. If you point at squares on the map, and tell the players, "this is difficult terrain," they are going to expect reduced movement speed, and that's it. If the GM then also imposes Dex checks without warning, the GM would be violating players' expectations about the rules, which isn't what I'd call recommended procedure.
Making a specific skill challenge out of normal stairs for one character... seems kind of petty simulationism of questionable quality.
A quick bit of searching of youtube turns up several videos of horses climbing stairs. It is a little awkward, but it is not clear if that's because horses really aren't that bright, aren't used to it, or (very likely, imho) that horses eyes are not placed so they can actually see what's at their feet very well. None of those should hold for a centaur.
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