This is a very interesting post - we had the same issue as to whether to choose public or private. I haven't read all the posts here so sorry if I cover anything that's already been said.
My son will start in a private school in Zurich, next August (we don't move to Zurich until end of June) for the following reasons. Note that the first two reasons were the most important for us.
1. Our son is very much a verbal kid. He's been accurately described as having verbal diarrhoea. He has struggled to learn a new language for the last 2 years (mandarin which, granted, is a very difficult language to learn but he seems to do worse than his class mates). So to put him in a local school where he has to learn both high german and swiss german from nothing, and where no-one speaks english, seemed too harsh. Our son also specifically asked not to go to a school where they don't speak English.
2. Our view of the local system (right or wrong) is that it is very regimented and authoritarian. We have always tried to encourage intellectual curiosity and therefore the IB program (which uses inquiry based learning) is a more natural fit.
3. Private schools usually have a lower teacher-student ratio which should ensure more attention to each child. That also helps if your child struggles with any particular subject as they should get more one on one. Kids from private schools also seem to be ahead academically of their public school peers.
4. Private schools in theory attract better and more motivated teachers. This is certainly true in countries such as the US, UK, Aus, etc, as most teachers would prefer to be in the private system than the public partly as their salaries are higher and partly as the students are generally more motivated - not sure if this is true of CH.
5. The parents themselves are generally better educated and as they have the option of choosing a private school, they tend to be more vocal and demand more of the school, otherwise they move their kids.
6. Similarly, as the parents from a private school are generally better educated their children are in a home environment where academic performance of a higher level is the norm. (This may also be true of the better public schools in CH.)
7. The facilities, resources, and opportunities for different subjects are better in private schools than most public schools.
8. The education from a private school is more "transportable" to other countries. It provides a more rounded education. Also (if you believe the private schools), the kids themselves are more adaptable because of the teaching approach - it supposedly produces children that are more independent thinkers and problem solvers. And because they're around kids from all corners of the world they're more exposed to different cultures and more accepting of differences (all in theory).
Having said all that, I think there are two major drawbacks with private schools (apart from the cost).
First is that you don't integrate into the community, don't learn the language as quickly, and the kids do have to deal with classmates moving on regularly. I think this is a big negative if you're planning on staying in CH long term (which we are).
Secondly, and I hope this isn't true of CH, is the expat brat phenomenon. We have seen this in HK - some of these kids have alot of money and seem to have way too much freedom. They are very arrogant and rude, and have no respect for other people. This is not being posh - these are intelligent but very badly behaved kids. While these kids are the exception, you would not want your child associating with them or being the butt of their pranks. (Thankfully our kids are too young for this at the moment.)
Sorry if I rambled - we went through alot of heartache trying to decide. Hope these thoughts help.