Green wood for chairmaking was part of a whole sustainable system of work, on a huge scale. Bear in mind that we're talking about "stick chairs" with slim round legs and woven seats, or "Windsor" chairs with stick legs fixed into holes in a shaped plank seat - a set of mortice and tenoned dining room chairs is a totally different proposition.
The components of a stick chair could be made from coppiced timber - using the thin, new growth of slender poles from the base of a cut-back tree. So there was no need to fell mature trees, just to manage the woods in a pattern of cutting back and harvesting. These slim poles needed no sawing. Thicker poles could be cleft - much quicker than sawing and producing parts of maximum strength, as the grain stays continuous through the whole length of each part.
The work of cutting and shaping could be done with a very few hand tools. (Green timber is much softer to work.) So much of it was done in the woods, where the trees grew - saving on transport. Seasoning was done, after initial shaping, and could be quite quick for the small sections used.
The design of a stick chair, with turned tenons fitting into bored holes, is very strong. As has been said, shrinkage can be exploited to make glueless joints that tighten over time. But the lightweight structure can also give a bit, to accommodate unevenness in the components, or a bumpy floor.