Yes, public schools can be excellent, especially in areas where high real estate values rake in tons of property taxes. The Cupertino/Palo Alto schools I listed are all public. Housing prices in the are are ~$800k or more.
There's a difference between snobbery and bullying/hooliganism. It's true that private schools probably produce more people who are snobbish, but these schools don't tolerate students who act up. I went to a Catholic school (as an atheist, and back when my family was poor). Plenty of snooty and sanctimonious religious kids. But you know what? The school tolerated no fights, and no harassment. And if you missed a homework or did bad on a test, they were on the phone to your parents the next day. As opposed to the public schools I would have gone to (if my mother hadn't sent me to private parochial school) which had regular gang violence, kids being SHOT for Nike tennis shoes, jackets, et al.
Public schools are a crap shoot. In Silicon Valley, the best ones are pretty much "private" in the sense that you pay a huge real estate fee to live in the best school districts in order to qualify. >50% Asian population in the schools. Parents in these schools don't tolerate their kids being idiots. But in less wealthy areas? The statistics work against you. The majority of the kids will be fine, and the majority of the parents will be fine, but you'll be much more likely to encounter some ruffians whose parents don't give a shit, and a school administration that is either a) apathetic to your concerns b) afraid of the students c) afraid of parents suing over any punitive measures against their kids d) already overloaded with so much problems from illprepared kids that they can't do anything.
The simple fact is, schools that require a higher amount of sacrifice by the parents are likely to have favorable demographics and parents who are much more likely to be in control of their kids and responsive to bad behavior, and much less likely to be apathetic. These schools also are much less tolerant of disruptive students. In fact, one of the K-8 schools I am trying to get my son into "interviews" the kid when they are 4-5 years old to see how disruptive they are. They reject kids who show tendencies to be disruptive and disobedient. Yep, my own son might be rejected if he fails his grace period. Basically, there is no bullying, fighting, or harassment allowed in some of these schools, any of those activities are punished and there is always the threat of expulsion.
In high school, I went to an engineering magnet school that treated school like work. There was no "school uniform" but you had to dress "business casual". They told us in the 9th grade orientation to treat coming to school like coming to work. That they would treat us with respect and allow freedoms, as long as we showed respect for each other and the school, as well as the ability to hand responsibility. People had to refer to one another by last name, and in many cases, with honorifics, especially school staff. Teachers were addressed as Sir. You had to keep your workspace, desk, et al, reasonably clean as well as the school grounds. Conflicts were encouraged to be resolved by talking, but if not, go to your teacher or principal and talk about it.
Students were collectively punished, so when someone acted like a jerk, everyone loss privileges or had to do extra work or time. This lead to less of a tolerance for people starting fights or picking on other people directly. Basically, people just formed cliques and talked behind people's backs, but there was no fighting, and no outward verbal harassment. Some guy tried to harass me once and a whole group of people, not my friends, rose up to defend me and tell the guy to quit the bs.
Was it ideal? No. But the current anarchy by which US high schools are run as "lord of the flies" fiefdoms and the assumption that somehow being subjected to harassment and bullying and having dignity stripped away "makes you stronger" is ridiculous. Whenever I talk to immigrants about high schools in their home country, I literally can't believe it. The level of bullying and infighting between tribes seems far less.
Essentially, I want my son to go to a school that thinks the primary purpose is learning and not "trial by fire socialization" Yes, socialization is VERY important to success in career/business, but I don't think social skills are best built by enduring hazing rituals, fist fights, and group heckling and harassment. There are lots of people who seem to think "well, we all went through that, and it's what made us successful. It makes you a man, strong." Personally, I don't think abuse, humiliation, threat of physical violence, is the mechanism to learn conflict resolution, how to influence people and make friends, and how to compromise and socialize.
I personally have a low opinion of most public schools on average, and it highly depends on the district. Atleast the elite schools and private schools lay out their vision and how they plan to run things up front. Often with a local school, you have no idea what their overall student population managerial style is.