Your pixel shader implementation is quite clean and simple. But the method I described doesn't rely on shaders at all, it uses the fixed function triangle raste

Your pixel shader implementation is quite clean and simple. But the method I described doesn't rely on shaders at all, it uses the fixed function triangle rasterization and depth testing. It has different performance characteristics. That method is also of historical interest because it works (and is fast) also on pre-shader graphics accelerators.