well, what you're calculating is not curvature but some curvature-indicating value. this, of course, makes sense since you're relying on normals which are per definition closely intertwined with curvature. but instead of building the covariance matrix you could just do pretty much the same by averaging the normalized normals from surrounding points and calculate the difference to the normal of the current point. however, from the mathematical and geometrical side of things neither this, nor your method has much to do with measuring curvature and the result is most probably off by a large number. (in your case, for instance, it's a min. and max. value of 0.000332 / 0.7385 compared to mean curvature of -107.395 / 204.042).
if you want to compute curvature by using eigenvalues you need to compute, for example, the shape operator by computing the jacobian of N, which in this case results in the hessian and finally project it onto the surface. building the covariance matrix of normals doesn't make much sense if you're talking about accurate curvature values.
building the shape operator is one of the methods described at houdiniGubbins and no, it doesn't need to be a triangle mesh. fitting a polynomial and computing curvature analytically also doesn't need triangles. in fact it doesn't need any mesh at all and works even on a pointcloud. the same is true if you're computing the curvature tensor based on a simple diffusion distance and search for k1 and k2 in the least squares sense. the most accurate results, however, are possible on a triangular mesh.
you're right that curvature computation on meshes is generally problematic but the reason is not because of irregular distances in the one-ring. the main problem is that second order derivatives of piecewise continuous functions are pretty much everywhere null and hence curvature is constant on a triangle. using the edgeTransportSop does not solve this problem. if we are talking about the smooth case, then your idea might makes some sense but on a smooth surface you simply wouldn't need it because here it's easy to compute derivatives. beside that, you would need geodesic distances and not the shortest path distances which the edgeTransportSop calculates...