'New Zealander' isn't really demonym, since it's a noun rather than an adjective. 'New Zealand' is both the name of the country and the demonym:
'The Russian prime minister' -> 'The New Zealand prime minister' (but you can't say 'The Russia prime minister')
The New Yorker mayor also doesn't sound right, although that's the correct demonym according to Wikipedia.
Would it be possible that perhaps the demonym is the name of the people, a noun, while the adjective is another thing, that can happen to coincide with either the place name or the demonym or something else, on a case by case basis?
