Question about time-complexity for MST-like algorithm
This problem looks variant of TSP problem(more precisely metric TSP). As you want to traverse all node with the minimum distance.

TSP is one of the classical NP-Hard problem.

What your friend did is a Greedy approach making minimum Spanning tree Which is 2-Approx(metric-TSP) of the minimum distance.

This greedy approach generally works in E log V time but does not have guarantee for exact solution.

By using graph structure we can implement in linear time also.