I use spokeshaves. Veritas is the brand I use.

I recently did some work in 16/4 cherry with both convex and concave curves.
It had a little tighter radii than you chairs but I experienced the same
problem.

I'll second the Veritas shave idea. IME, shaves to a surprisingly good job
of ensuring a smooth curve (not to be confused with a smooth surface) which
I think is what is eluding you. You would think that you need all of that
sole of a compass plane to control the curve, but I'm not convinced its
necessary. The Veritas shave with a convex sole, is very much like a small
compass plane and it performed well.

Another approach that occurred to me however:

The problem with the OSS is that the drum diameter is too small. If you have
a lathe, you could fabricate an 8" or 10" diameter sanding drum. Although
the surface speed might be but crazy. Just a thought.

One other thought on the abrasive approach: Start with the biggest and
nastiest grit you can find. My initial attempts with the 3" OSS were with
medium grit. It tended to ~divot~ because it was too tempting to apply more
than the lightest pressure. A light touch helps achieve the smooth
continuous motion you need with the OSS.

In the end, I used a combination of shaves and abrasives, shaves really
shine when smoothing (shaping) the curves. My technique is not good enough
for a final surface, so the OSS was effective at smoothing the surface (but
I did have to work through the grits)

-Steve