My research adviser used to be on the REU selection board for Harvard SAO's REU program. He said there's lots of elements of randomness that play in: things like how well suited a particular student would be to work for such and such a mentor, if that mentor was looking for someone with specific skills to do something like...say...computational physics, or if they wanted anyone and everyone.
I applied to 13 REUs and got rejected to all but 1, which wasn't even an NSF REU at all (REU equivalent for University College London). I don't feel cheated, but rather lucky and extremely grateful. I have to live in the shadow of my class valedictorian who does research with my adviser all the time too, so I know how it goes as far as feeling unappreciated goes, but remember that REUs are extremely competitive.
Going to an Ivy League school does not automatically qualify you to get into an REU program. Me and my valedictorian friend are the only 2 in our class to get accepted for summer research internships. He got accepted to Kitt Peak, Harvard and one other (he picked Harvard, of course), but we both go to a little-known state university, are white, etc. What sets us apart from other students in our dept. is that our adviser has given us challenging, unique research projects to work on and we've done lots of extracurricular stuff as well.
Sometimes REUs don't take those into account, but then again, that falls into the factors of randomness category. If you are a victim of aiming too high, perhaps, a better strategy would've been to go for REU positions where they have a large number of positions to fill (DOE, SARA programs), aren't page 1 research schools, or even hunt for summer research internships over seas. Canada is very close by and has some fantastic institutions to do research at!
If you're going to apply for an REU position in abstract algebra, you're going to be required to know abstract algebra to do it. Plain and simple. REUs teach you how to do research, yeah, but they don't teach you the material you're already supposed to know. If you didn't know it, why did you apply for [x] position in the first place?
Stats I've run from numbers I've gotten mailed to me point to the fact that upwards of 200 people fight for some 8 or 10 positions at major institutions. Suggesting to interview all 200 of those people is going to be time consuming at the very least.
I don't see why you're so distraught. When I got an opportunity to do research with my adviser this last summer, I was ecstatic, despite taking classes at the same time. In fact, through the summer and fall quarters, I wasn't even paid. I didn't do it for the money, I did it for my love of astronomy. If you've landed a research gig with someone at your dept. why worry? If it's because of the money, get used to it: scientists don't make as much as entertainers.