We present a new strongly polynomial algorithm for generalized flow maximization that is significantly simpler and faster than the previous strongly polynomial algorithm [34]. For the uncapacitated problem formulation, the complexity bound O(mn(m+n log n)log (n²/m)) improves on the previous estimate by almost a factor O(n²). Even for small numerical parameter values, our running time bound is comparable to the best weakly polynomial algorithms. The key new technical idea is relaxing the primal feasibility conditions. This allows us to work almost exclusively with integral flows, in contrast to all previous algorithms for the problem.