Do rich kids have an automatic advantage in college admissions?
No, not for the schools that matter. Mid-level privates will give an edge to full-pays, and will court wealthy families as potential future donors. But, the money alone really doesn't mean a thing to sweatshirt schools with massive endowments.
I've found that the further you get into the top 25 schools in the country, the less inclined schools are to look at bank accounts. What I have found, though, is that these schools look for is connections. Who's important in China? Who has a father high up in the Communist Party, where we could use a personal connection to get a foot in the door in a heretofore closed-off province? These are certainly not questions at the front of anyone's mind when reading an application for the first time, but these connections can ultimately play a role - especially if they're significant enough.
This all said, I think that colleges are just as inclined (if not more so) to give an advantage to low-income students. Of course, there are a ton of things about college admissions that are sketchy, but you can't impugn the motives of most people in the industry; people in admissions are almost all true educators. They love to hear about students with commitment, passion, overcoming obstacles, and so forth. The Dean and Provost might push for connections, but the worker bees in the office will always be pulling for the interesting kid who's had to work and fight for what they have.