A plain upholstered back is harder than a buttoned one, and this chair is a good demonstration of why.
Buttons are forgiving. Every button pulls surplus material into a hollow, so a cover that is slightly too big simply disappears into the pattern. On a plain tight back there is nowhere for surplus to go. Worse, this back is arched at the top, and a flat piece of cloth pulled over a curve always has more material at the outside of the curve than it needs - which wants to gather into soft ripples right where the eye rests.
The solution is to remove that surplus rather than to stretch it away. Small wedges are cut out of the cover at the tacking edge, spaced around the curve, and the cut edges pulled together as the cover is fixed down. Each wedge takes out a precise amount of fullness, so the face of the velvet comes up flat and smooth with no tension marks radiating from the corners.
All of that happens under the carved gilded border, where nothing is visible afterwards. What you see is a plain arched panel that looks effortless - and that could not have been achieved by pulling harder.