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.
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)
In any case, kindergartens are not entirely free. If the kids have to stay after 12.00 it gets very expensive.