It seems like people with medals in International Science Olympiads (maths, chemistry etc.) all have close to 100% chances of getting into Ivy League Schools. Are there even any more places for the rest of us?
Yes, there are definitely New Zealanders every year who get into Ivy League without an olympiad medal. An international-level award certainly helps you stand out in your application, but it is only one portion of a holistic review process. Do not be deterred from applying to Ivies for this reason alone.