Approximate max weight path in directed graph
You can't.  That approach is a dead end.  Finding the maximum (simple) path is NP-hard, by reduction from Hamiltonian cycle.  See, e.g., How to prove NP-hardness of a longest-path problem?, How is the Longest Path Problem NP complete?, Longest cycle in a digraph, https://en.wikipedia.org/wiki/Longest_path_problem.  It is also known to be hard to approximate, so you shouldn't expect any such simple approach to give a good approximation for all graphs.