This is a tangential point to your main question but has a bearing on my suggested solution:
I have heard that the old-school way of creating a flat surface is to use a hand plane, and that if you use the right tools, you may not even have to sand because the surface is so smooth.
While that can be true it does tend to get overstated by some planing enthusiasts. It was actually quite common in the past for even the surface left by a "smoother", or smoothing plane, to be further refined. Be it by scraping and/or with some abrasives (even before the advent of sandpaper).
Can I use a flat hand plane to smooth an flatten a curved surface?
The surface in mind is a U shaped stool top, so there is only a single curve
So I'm presuming this is dished, i.e. has a concave surface. Generally speaking a regular flat-soled plane can only be used on convex surfaces, not on concavities because the flat bottom will 'bridge' from toe to heel and prevent the cutting edge from contacting the wood.
There are a number of other edged tools that can be use for concave surfaces. In terms of smoothing the surface specifically (rather than creating it in the first place) the main one is the compass plane.
The all-metal version had a flexible steel sole whose curvature could be adjusted to match the surface you were working on:
Wooden versions are also an option and these can have soles that curve in one or both directions, the latter allowing work on compound curves (a surface dished both front-to-back as well as side-to-side).
Compass planes are specialised pieces of kit and can be quite expensive, but are of course purpose-made for the job and do it well and efficiently.
There is a simpler option, one much better suited if only a small amount of smoothing is needed and that is the curved card scraper. Card scrapers are widely sold in sets of three shapes, and this may be the ideal way to buy them as all three shapes will prove useful. For use on a hollow seat the best choice is the one that looks a little like a stylised comma:
All card scrapers can be used in either a pulling or pushing motion depending on user preference and the demands of the shape being worked.
After scraping you may end up with a surface you deem finish-ready, but this isn't always possible (due to vagaries in the wood as well as how sharp the scraper is) so don't feel any embarrassment if you need to do final smoothing using sandpaper.