I moved away from Männedorf many years ago, so can't help personally - but back in the day, most of the young parents I knew were involved in the Eltern/Kind or Eltern/Baby clubs:
https://www.maennedorf.ch/jugendfamilie/32068
This will be officially German speaking, but (again, back in the day) lots of folks spoke English. Don't let language keep you from joining local activities, you'll likely find that even a few sentences in German and a friendly attitude will open doors. Typically Eltern/KInd type clubs are great for networking - for finding babysitters, arranging play dates for toddlers, and of course for sharing a cuppa with other parents.
There are also sports clubs, an indoor pool, and of course the Badi come summer. You'll likely find that any place with activities for children will bring opportunities to connect with other parents.
Browse the village website above, see if there is any other club or cultural offering that interests you. People seem to be most open to new connection through activities.
One time honored way to spend a Sunday seemed to be walking the Pfannestiel as a family - that's how we met several of our friends.
Männedorf is a lovely village, and I thoroughly enjoyed my time there. I hope you do as well.
All the best.
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