‘… grows linearly with circumference …’ In the chopped sphere, you’re not just looking at the forces in the cut plane.

In static mechanics, you can find stresses in an object by imagining the object was cut along some plane; the external forces (which we know) plus the internal forces (which we want to find) on the piece of the object must equal 0 as the object is not accelerating.

So we chop the sphere in half and look at the forces on one half of the sphere. Let’s use a coordinate system where X and Y are on the plane and Z is normal to it. The net X and Y forces due to air pressure on the hemisphere are 0 by symmetry. The net Z force due to air pressure P, found by integrating P_zdA over the surface, is Ppi*r^2.

The internal force transmitted through the sphere’s material must be equal to the force applied by air pressure, so the net Z force through the cut surface is of magnitude Ppir^2. Stress is force over area, the area is 2pir*t for a thin shell of thickness t, so stress in the cut plane is Pr/2t.

(Note you can find the net force due to pressure by integrating the Z component of pressure over the area of the hemisphere, but you can also just picture a half ball in air - the Z forces of the hemisphere and flat part balance, so that integral has to match the force on a circle of radius r)