What should I do if the admission committee has missed something in my application and rejected me?
I am assuming you don't have a prospective advisor at this institution working with you on your admission. If you do, they will be in a much better position than us to tell you how to navigate this situation!

The concrete policy would depend on the country or individual institution, but to me it seems unlikely that they would reverse such a decision after the fact. In all probability they have limited places that they have filled already. They might also have rejected you for reasons they don't like to say openly, such as because the other candidates were equally good and didn't require double-checking the paperwork.

To know more about your chances of appeal, you would have to look into how similar cases have been handled by this department in the past, but that might be quite difficult without a connection to someone with historical insider info. While you are contesting the decision, the administration will naturally try to say as little as possible to avoid giving you ammunition.

If re-applying next year is an option for you, I would try to strike a more collaborative tone and ask how you should present your documents so as to avoid confusion in the future. And also try to strike up some contact with a faculty member who might help you. They don't necessarily need to actually do much, but an internal interest will give the administration some incentive to spend time answering your questions.