As we are traveling from Switzerland, I assume that child needs to travel on CH passport, and wife and me on EU passport. Is that correct?
We are planning to travel to US as a family and need to apply for ESTA. Me, wife and child all have EU passports. Child also has a Swiss passport.
I don't think Switzerland cares on which passport you exit the country, though I might not be right. The US doesn't care, as long as you are not a US citizen - if you are, you need to enter on a US passport.
Doesn't matter when entering USA. However, your child needs to enter Switzerland on their Swiss passport and as you want the exit and entry stamps of USA in same passport, it would be much less complicated for them to do the whole trip on the Swiss passport.
Maybe looking at it through the pedantic goggles of the US border staff, they might wonder why your child has a different passport to the parents? Perhaps take along the birth certificate and any other documentation, just as belt and braces.
We'll definitely bring both passports, just to be sure.
Don't forget your drivers license for the rental car.
You go in the foreign line, irrespective of your non-US family mix - all requiring ESTA