^ Perhaps I am reading it wrong. There's a difference between checking every value (which I agree you'll have to do) and checking every subsequence (which you indeed do not have to do) in the MSS problem.
I agree that you will have to check every value in the maximum subrectangle sum problem, but I am hesitant to agree that you will have to check every subrectangle, for the same reason. It's possible you're right when you say you'll have to check every subrectangle, but you might be able to get away with less than O(n^2). I don't know. I apologize if I'm reading your post incorrectly.