Imagine you have two blackbody radiators with the same bulk properties, except one has surface area shenanigans like aerogels. In the far field as a whole, it seems like both should radiate essentially the same regardless of the internal details. You can shape emissive direction, or improve efficiency of non-ideal materials, but even ideal materials don't fix the issues pointed out by the parent.

Right; it's only area exposed to the exterior that counts. A physical object can't thermally radiate more power than a perfect blackbody spanning its convex hull.

(This follows because a physical object can't absorb more light than a perfect blackbody spanning its convex hull. A perfect blackbody by definition absorbs 100% of incident light, which is a hard upper bound. Any line incident on an object is also incident on its convex hull).

(Consider an isothermal object that emits more power than a blackbody in the shape of its hull at the same temperature. If you were to place that object in a closed system at thermal equilibrium, the interior of an insulated emissive sphere—combining assumptions, it would emit more power than it absorbs, in violation of the 2nd law. Starting from an isothermal system, the object would grow colder, and the enclosing container hotter).