I can tell you that my SOP and my entire process was fumbled and poorly planned.
My GPA is a 3.822, my GRE is ~90th+ percentile across the board, with subject at ~75th percentile. But when I heard that the biggest factors were research, letters, and SOP, I was sure I was screwed. I have a summer of research. 8 weeks. Maybe 8 weeks and 6 days if you count the MCM. I have no papers that are submitted, let alone published papers. None of my SOPs were really tailored to the place where I was applying. My lack of specific research interests makes me look like a generic candidate. None of my letter writers are big names. And I'm pretty sure I pissed off one of my letter writers in the process of applying before she submitted anything, so what she submitted may have ended up altered.
But. I got in somewhere (incidentally, since the application was due there first, that is where my application looked the worst). And I'm still waiting to hear from other places.
I will tell you though that there is absolutely no one that gets in without the requirements and qualifications. I mean, come on, many of us are basically asking these places to pay us to go to school. And almost all of us are asking for the opportunity to work closely with some of the greatest minds in our respective fields. They've gotta be selective. That said, from what you've shown here, it doesn't sound like you're out of the running (of course, it all depends on how well you fit in the places you applied, where you applied, and how good your entire profile looks, etc).
The application season ain't over 'til that last letter rolls in.