Remember the backgrounds of the kids.
Regarding what you say about the characters being priveleged, we need to remember that that is the background of those kids at home. While we don't want to indulge kids too much while they're at school, we also don't want to make their school too much different from what they're used to at home, particularly at a boarding school. If we do, kids will get homesick and become maladjusted. Therefore, I am less concerned or disturbed about the privileged lives of these kids than I normally would be. You also pointed out that there's not much adult supervision. True, but the kids are well-behaved. While you could argue for more adult supervision, nevertheless the kids show that they can be trusted. We need to remember as well that this is a boarding school. At those places kids are on their own a good deal of the time. There are the standard messages about friendship, etc. What makes this one stand out is that the title character, despite her popularity, is nice to everyone she meets. (Maybe that's why she became popular in the first place.) On many shows, by contrast, the most popular kids tend to be stuck up. And, while they may or may not be at the top of their class, they usually do well in school and do not empathize with kids who struggle in school. Zoey does. Zoey and her friends are also willing and eager to show the boys that girls can do anything that boys can. Indeed, casual viewers would never know that this had been an all-boys school until Zoey's class started. Lastly, although classes are rarely shown, as most episodes take place in the after school hours, the few times we do see classes we get the idea that they're reasonably challenging, thereby discrediting the notion that life is easy there. The show is even made better by the terrific acting by Jamie Lynn Spears. She's even better than her older sister Britney (although Britney is good too).