Embroidery and upholstery pull against each other, and if you stretch them together one of them loses.
A plain cover wants to be pulled tight over its padding - that is what gives an upholstered back its shape. An embroidered panel wants the opposite: stretched hard, the ground cloth distorts around every worked area, because the embroidery itself does not stretch. The flowers pucker, the ground goes wavy between them, and a piece of work that took days looks careless.
So the two are not stretched together. The embroidered panel is mounted on its own light sub-frame, tensioned only as much as it needs to sit flat - which is considerably less than upholstery tension - and that sub-frame is then set into the chair back as a unit. The surrounding upholstery is pulled to its own proper tension around it, and the two never fight.
The panel is held in place rather than fixed permanently, which means it can be lifted out. An embroidered panel is the part of the chair most likely to be treasured and least likely to survive a spill, and being able to remove one for cleaning - or replace one entirely without touching the rest of the chair - is worth building in from the start.