I haven't checked your replacement in detail, but a working example can be found at Discrete_Hilbert_transforms, if that's what you are looking for.
There are two key concepts in the article.  Fully half of the main section, and all of the math, is devoted to the concept depicted in the figure, which is that of doing piece-wise convolution and creating a seamless result.  One has to understand that before the next step makes sense, and it is easier to understand without the unnecessary complexity of circular convolution.  Divide and conquer.  It's about teaching and understanding, not efficiency.
The details of circular convolution aren't included here, because they are explained in other articles.