I second your thoughts. Our application was turned down and I got the same email in my inbox this morning. In my case, I had a different reaction to the rejection though, in that, I decided to take on the challenge head-on. But then again, now that I think of it, I'm deeply devastated that I couldn't get any feedback from them.
I'm a very curious person and have been all my life. Part of what makes me unique is my ability to go to great, if not seemingly impossible, lengths to find out things that pique my interest. And, YC, finally seemed like a place where I could belong after feeling so very different all my life.
It is painful to see them return your application with a generic 'no' because I have no way of knowing what I could improve or what is it that made them look away in the first place. I'm pissed off because I care.
To quote from Don Quixote, "there were no embraces, because where there is great love, there is often little display of it."
I hope they could give me some feedback. For once in my life, I'd know, for a fact, than having to ruminate and speculate over it.
"Why not" rings very true and can be taken as encouragement to soldier on. Yes, it would have been great to have YC's backing, but at this point it's a bit of a lottery, isn't it? And if your project cannot succeed without YC, that is not a good sign. Once more into the breach.
I'm sure that if you posted your app/site here, people could give you plenty of reasons. Ask yourself what you think the reasons are. What would you do to "kick your own ass"?
Technically, I am not buying the "there is no reason" either most of the time. It is probably any number of common reasons like red flags (filled with buzzwords and marketing speak, weak/iterative product, single founder with no history of success) or maybe it is just a bunch of more minor or subtle things. Maybe they don't think there is a huge future for X even though it is popular now; they think Y is on the way up or there are just 10 other applications doing something like X which are more compelling or have a team with a better pedigree than you.
Maybe sometimes it is fairly easy to give a reason, but then it will be expected for everyone and they get a lot of applications now. It would be hugely time consuming to give a detailed evaluation and maybe sometimes too it would be giving away some secrets they don't want to.