It seems to me that routing plays a central role in almost every area of computer networking. It is such a central part of networking that I suppose it is fair to say that routing is the core of networking. Certainly routers are at the physical core of networks.
Transport is distinct from routing. Our study of transport centered on congestion control. Congestion happens at routers. Congestion is essentially a clogged route. The most effective congestion control schemes make use of explicit congestion feedback from routers.
Much of what we studied in the application space had to do with locating distributed content, which can also be seen as a routing problem. Perhaps this is a stretch, but the big picture is that applications are being used to find a better path for content delivery.
Our study of Internet architecture focused on naming and addressing. Names and addresses are for routing. Our study of wireless has been dominated by routing challenges specific to the wireless environment.
We started this class with the end-to-end principle. What is the middle? It is the network. What is a network? Isn't it essentially routing?