Under what circumstances is the Aircraft 'Stability-Axis' system used in either design or analysis of aircraft?
Body-axis is used to construct the equations of motion and express the aircraft angular velocity and Euler attitudes. By conventional, it is defined such that the x-axis is aligned with the nose and the x-z plane is the plane of symmetry.

Stability-axis, whose x-axis is aligned with the longitudinal incidence of the flow against the body-axis, is used to express aerodynamic data, which are derived from semi-analytical methods (e.g. DATCOM, ESDU), computational methods (e.g. vortex-lattice methods, Navier-Stokes), wind tunnel testing, or full flight data. 

Stability-axis is used for aerodynamics because lift and drag are conveniently aligned with the axes. And as the OP alluded to, it is much easier to rotate the aircraft against the wind tunnel balances, than to rotate the flow against the test model.

Wind-axis has its x-axis aligned exactly with the flow velocity vector. It's not really used anywhere. We don't use wind axis because it breaks the x-z symmetry and makes things more complicated than need be. It does not offer any additional benefit than stability-axis.



Image from: https://www.mathematica-journal.com/issue/v8i2/features/codegeneration/contents/html/Links/index_lnk_3.html