Have Waterloo updated their site recently?
They have a "Spreading Resistance of Circular Source on Circular Disk with Edge Cooling" which I have never seen before. It appears to generate almost zero spreading resistance with any kind of convection on the edges. This is the closest I have seen to simulating a hemisphere. I think that for the hemisphere spreading resistance might actually
be close to zero, it's a radial thermal gradient of constant L.
This page is also useful.
My biggest problem is to figure out what is going on in the area just above the die at radii less than the die size, I am at the moment assuming a hemispherical surface of the same area as the die as a start point for the onion layers. Squashing this flat would give a similar geometry to the actual block but I know it's not quite right.
Share your unhappiness about the 45 degree thing, there are more accurate ways. Simple though for illustrating the idea and maintaining continuity of calculation method (slices and shells).
Edit: I think I'm leaning towards your way (skipping 45deg) . I'll recalculate.