As was implied in another post, it doesn’t quite work that way. Schools like Vandy get more than their fair share of what I guess I would be forced to call “deluded dreamers”, people that apply to schools like Vandy Duke, Ivies, etc. with ACT scores of 25 and a GPA of 3.25. Barring being a recruited athlete or a descendent of the Commodore himself, they are not getting in but they are counted in the 87% denial rate. Clearly your chances are better than theirs, ergo your chances are better than 13%. I would say in your case 20-25% is probably accurate, but it might be higher even. We cannot know some aspects of your application that the adcoms will see.
As far as essays, I think you might not be looking at them quite the right way. Suffice it to say they are in fact more important than you might think, beyond any “weighting” you might have been told. Sure, if you have stats that are well above the average for a certain school the essays might not matter, but even that is not always the case. Many schools use the essay to see if that student is really interested in attending, or if they are far more likely to go elsewhere and is just using the school as a safety. And certainly among the highly selective schools like Vandy, once they have it narrowed down to a set of students that are very similar in their profile, the essays might well be the tie-breakers. At that point any weightings you were told or read about are out the window; the essays become all important. The bottom line is that all schools are different, and it isn’t always how it appears. Tulane, for example, calls their “Why Tulane” essay optional, but it is only optional if you really don’t care about getting in.