Find the minimum edge size, L, that would allow the construction of a tree whose edges connect all given 2-d points, but no edge exceeds length L
Yes, your algorithm is correct. The correctness can be proven by induction on $S$ easily.

Instead of growing by selecting the nearest point to $S$, a faster way in general should be just selecting the shortest edge available and including both of its end points. How to select the next shortest edge available? Construct a min-heap of all edges up front. Pop out the top of the heap one by one. Using a counter you can stop once all points are included. The length of the last edge included is the minimum edge size wanted.