Computing the most central nodes in large-scale commodity networks is rather important for improving routing and associated applications. In this paper, we introduce a novel framework for the analysis and efficient computation of routing path-based centrality measures, focusing on betweenness and traffic load centrality. The proposed framework enables efficient approximation and in special cases accurate computation of the aforementioned measures in large-scale complex networks, as well as improving/adapting commodity (traffic) routing by identifying and alleviating key congestion points. It capitalizes on network embedding in hyperbolic space and exploits properties of greedy routing over hyperbolic coordinates. We show the computational benefits and approximation precision of our approach by comparing it with state-of-the-art path centrality computation techniques. We demonstrate its applicability on real topologies, characteristic of actual large-scale commodity networks, e.g., data, utility networks. Focusing on two graph embedding types, Rigel and greedy, we compare their impact on the performance of our framework. Then, we exemplify and statistically analyze the dynamic routing adaptation, via the variation of the minimum-depth spanning tree employed for greedy embedding in hyperbolic space. Notably, this allows for efficient routing adaptation according to a simple, distributed computation that can be applied during network operation to alleviate arising bottlenecks.