It seems nobody bothered to mention French Polish. Sigh. That is the standard piano finish for quality pianos, period.
French polish is a series of steps that fills and smooths surfaces using shellac.
Over a stained and filled (grain filler) prepared surface:
The first step uses rottenstone (like pumice) mixed with 1/2 lb or 1 lb cut shellac; rottenstone comes in grades (particle size). You work through grades of rottenstone + shellac from coarser to finest. The final multiple passes are shellac only.
The reason this is not used more often is not the quality of the result, but the cost of paying someone with experience to do it. Mass produced pianos would have to be finished in places where wages a far lower than Europe or The US in order to compete price-wise with other finishing methods.
I've touched up some baby grands and estimate that one square yard of surface is about two hours of work for minor touch up. I do not know firsthand, but am told a full grand piano can take 160 man hours or more for all finishing steps.
The good part about french polish is that it can be touched up to look completely new, but damage on nitrocellulose lacquer may require complete refinishing. By damage I mean spills, not where little Tommy takes a hatchet to a piece of furniture.
Pictures do the best job of conveying the idea: