Re: adcoms: it's not just about how brilliant you seem to them. If they have a choice between two excellent candidates in an area, they might go with the one who seems more collegial to them, or the one who seems like a better fit with the other members of the new cohort. These adcoms are selecting people they'll be with for the better part of 6 years, and they're trying to create what they think is a well-balanced cohort. You might be a genius, but if you're a genius with an ego and it shows, that will get you rejected at some places. In other cases, adcoms will have decided to introduce different areas of interest into their program, and that might mean they select students with those interests even if they do not already seem like a polished genius. Also, lots of programs look for potential rather than genius or highly developed professional skills (and if you publish early, you're stuck with that publication for the rest of your life, and you might regret it one day).
Basically, the quality of your application and all the accompanying details matter, but whether you get in or not cannot be reduced to how outstanding this or that application is.
Other factors matter to them, too, and you can't guess what those are.