I believe they used custom scrapers or scratch cards to finalize the surface. Concave curves of varying shapes are very difficult with planes, and would take many different ones - extremely time consuming especially for a one off piece. My thinking is they had to start with a shaped moulding and steam bend it.
Just a speculation. In those days hogging out the material would have been done with chisels and big gouges (and big mallets!). Profile gauges (shaped scrapers) to monitor progress.
Last time we were in Charleston we went through a couple museums I noticed the moulding on a secretary looked like it was bent. The reason I think it was bent was that sawing out a curved moulding (or serpentine drawer front, etc) exposes endgrain, which be difficult to deal with if staining. I could be wrong about that.
Cremona’s technique is pretty neat, but you can see how endgrain will be exposed. OTOH if it were a bent, I guess it would have to be one piece of wood, and that’s a massive amount of bending with the potential springback becoming an issue fitting to the case. I don’t know how a lamination would work.
From Glen Huey’s (excellent) book