How does Schwartz's paradox of surface area effect modelling of 3D objects?

When approximating the cylinder, for example, such an approximation does work, so long as we also have convergence of the normals.

The approximation fails for Schwarz lanterns because the normals of the triangles in the Schwarz lanterns don't converge at all, much less to the normals of the cylinder.

See for example here of how we can use triangulations to approximate surface area: http://arxiv.org/pdf/1404.1823v1.pdf

Also look at the last paragraph of the paper linked to in the original question -- Lebesgue already solved this problem in 1902 by demanding uniform convergence instead of just pointwise convergence. Clearly Schwarz's lanterns converge pointwise but not uniformly -- the reason why the surface area blows up is because the triangular approximations can become arbitrarily jagged (which is also why the normals fail to converge). With uniform convergence, this doesn't happen.

See this Wolfram demo for how it works: http://demonstrations.wolfram.com/CylinderAreaParadox/

If it is reasonable to demand uniform convergence of functions instead of just pointwise convergence, then it should also be reasonable for surface areas.