At Oerlikon station (on the main line between ZRH Airport and ZRH Hauptbanhof) just off platform 8 on Max-Frisch-Platz 25A (it's a van).
This testing location is part of corona-testing.ch which has other locations in and around Zurich.
You get a certificate, but if you want your passport linked to it the fee is 10 SFr extra--Delta doesn't require a PP link.
PS: I just tried a GOOGLE search "free covid test Zurich" and didn't find the site--
BUT "covid test zurich oerlikon max-frisch-platz" did.
PS(PS): I had an appointment, but in the 10 minutes between testing and getting the results I was the only person at their "doorstep"
Super helpful! There's other locations around Zurich depending where you'll be so, assuming the entry rules don't change, I'll be using this service myself in a few weeks.