The algebraic manipulation that you use to break the dependency seems to be specific to the matrix being Poisson on a finite-difference grid (is that right?). In that case wouldn’t it make more sense to compare against specialized Poisson solvers?
In any case, the idea behind this series on Jacobi (this is like the 4th post I believe) is to help students (and people interested in scientific computing in general) get what actually matters computationally and lift the hood to get some understanding of the interplay between such algorithms and their actual implementation on hardware.