Unless the culture is wildly different than here in the US I don't think it's reasonable to consider private school and nannies as part of "cost of living"... Those are both unnecessary luxuries.
In Zurich, there are no public kindergarten. The monthly fees ranges from 2.500 to 3K+ a month. A nanny is necessary, when both parents have a full time work and family is not around (like in my case).
I think what you are talking about are "Kindergrippen" (when kids are 2-3 years old). Kindergardens are public and free.
This Kindergrippen-age is really the only moment when kids in Switzerland cost you significantly more than in the rest of Europe.
Think it this way: From the actual kindergarden age (4 years old?) you won't pay anymore. It's even free to send the kid to the best technical university in mainland Europe, which is supposed to be ETH (think MIT of Europe)