Is this dense version of Kruskal's algorithm well-known?
I'm not sure about this specific method for achieving $O(n^2)$ time, but two different methods for performing Kruskal in $O(n^2)$ time are given in my paper "Fast hierarchical clustering and other applications of dynamic closest pairs" (SODA 1998, arXiv:cs.DS/9912014, and J. Experimental Algorithms 2000):


Use Prim–Dijkstra–Jarník instead and then sort the edges to get the insertion sequence that Kruskal would give, or
Use the quadtree closest-pair data structure described in the paper, viewing Kruskal as a standard agglomerative clustering procedure where we merge the closest two clusters into a supercluster at each step, with "closest" defined as the length of the shortest edge connecting two clusters.


Solution 2 is similar in spirit to what you describe but the details of how to keep track of the distances between clusters are slightly different. You keep the row-wise minima of the cluster distance matrix, allowing you to scan this list of row-minima in linear time to find the global minimum, while my paper overlays a quadtree on the same matrix and keeps track of the minimum in each quadtree square. Your method is simpler, but less flexible for some other dynamic closest pair problems (it depends on the fact that merging two clusters causes their distances to other clusters to decrease, true for this problem but not necessarily for others).

As I wrote in 2011 in the Wikipedia article on the nearest-neighbor chain algorithm, that algorithm can also be used to perform Kruskal in $O(n^2)$ time. However (unlike some other applications of the nearest-neighbor chain algorithm) you don't get a space savings, so (like the quadtree method and your method) the space is still $O(n^2)$. In contrast Prim+sorting can use only $O(n)$
space beyond that needed to store the input.