I'm a U.S. citizen through my parents, even though I was born in New Zealand, where I've lived most of my life. I came to the U.S. to visit, and realized I'd like to spend more time here. So I made an appointment at a U.S. passport office, figuring it was time to get proof that I'm a citizen. But they told me I had to come back with a Certificate of Citizenship. I looked at the instructions for Form N-600, to get the certificate, and it says that one of the types of evidence I can present is a U.S. passport! This seems very circular. What's up?
What's up is that you seemingly encountered a U.S. government official who hadn't read the details on how to do his or her job. If you are a U.S. citizen through your parents (as described in Nolo's articles on Acquiring or Deriving Citizenship Through Parents), you are supposed to have a choice of whether to apply for either a U.S. passport or a certificate of citizenship, or both, in whichever order you like. Many people choose the passport first, or only the passport. It's cheaper than the certificate, and ultimately more useful.
The probable source of the confusion in your case is that a U.S. certificate of citizenship is considered "primary evidence" of your eligibility for a passport, as described on the State Department's list for first-time applicants.
But in case those items aren't available, the State Department also offers a list of Secondary Evidence of U.S. Citizenship. And that list includes the type of documents you probably took with you to your passport appointment:
Had you known at the time, you could have asked to speak with a supervisor. At this point, it's probably best to make a new appointment. Or, if you're heading back to New Zealand soon, you can also apply for a passport through a U.S. consulate there. Maybe you'll have better luck with the consular staff, who are accustomed to meeting U.S. citizens born overseas.