I see what you're getting at -- but this isn't competing in the Ironman event or racing in the tour de France (instances where one person can overcome extreme odds and come out on top due solely to their own performance/aptitude).
As much as we, as applicants, like to think that we have our ish under control (i.e. I worked hard and got a X.XX GPA, I studied my ass off and earned a YZ mcat score, I did this and that and some of that too. My merits are now astronomical)... it isn't the case. The summation of our efforts essentially boil down to one (or a few people's) subjective opinion of our entirety in a couple paragraphs on a piece of paper. It's all up in the air and no matter how awesome one is, they still might not get admitted to their dream school because of subjective opinion of someone else.
With that said, I would be the counterpoint for your stance: in cases like this, I'd take advantage of my connections if I had them and never look back.