Optimal root in shortest path tree (SPT)
This problem is equivalent to computing the radius of a graph. The best known algorithms for computing radius essentially compute all pairs shortest paths, so the method that you suggest cannot be improved much. If the number of edges is large, maybe Floyd–Warshall algorithm could be faster.

References:

https://en.wikipedia.org/wiki/Distance_(graph_theory)

https://arxiv.org/abs/1506.01799