Both of you make great points as to the randomness of the application process, but I do think such points shouldn't undermine a general predictability concerning the way schools make decisions. Similar to how one can distinguish a grade A paper from a grade B paper, so too can one distinguish different types of applicants from the select pool. For e.g., what makes paper 1 an A paper might be its clarity coupled with an original response, whereas what makes paper 2 an A paper is its organization and its ability to draw upon interdisciplinary research to back up a thought that is not their own.
As for the details of the application; the point I was trying to make is that if what one would call a strong (questionable) profile (excluding the GRE scores) cannot getting into a T20, then what else is required? There seems to be something missing, and some of your suggestions have been fit, randomness and more. However, there seems to be reasonable grounds to question that such considerations are the leading factors. Even in this year's applicant pool, there have been individuals who have received multiple offers to many T10 programs. It doesn't seem random.
But the question I had initially asked is: what is your take on asking the programs in which one has not been offered admission, for their advice on how you can improve your application? Or what could you have done differently in order to ensure entry into the program?