“New Zealander” is perfectly normal and cromulent for this American. There’s nothing stilted about it, so far as I can tell. That, or “Kiwi,” is what you call a person from New Zealand.
Now, yeah, “Netherlander” does stick out a little.
Old Zeelander, then.
Heh, heh.
j
I personally prefer Schweizer. I expect many, if not most Schweizer do too.
Historically, it wasn’t always the case. The Oxford English Dictionary includes senses for “French”, “Irish”, and “Welsh” as count-noun demonyms, with plural forms “Frenches”, “Irishes”, and “Welshes”. That is, “French” has been used to mean “a Frenchman or Frenchwoman”, “Irish” for “an Irishman or Irishwoman”, etc. Quotations are provided from several well-known authors:
All of these senses are marked as obsolete in the dictionary. So the real question here isn’t why we don’t have count demonyms for these nations, but why the ones we do have have fallen out of use.
The OED lists singular demonyms “Chinese”, “Japanese”, “Lebanese”, and “Portuguese”, none of which are marked as obsolete. So in the assessment of at least some lexicographers, it’s perfectly cromulent these days to refer to “a Chinese I know” or “that Portuguese over there”.
I thought they preferred “Helvetian”.
Because historically the male formation has been the gender-neutral variant, or rather than masculine form also serves as the neuter form in most context in English. There are plenty of non-English neuter descriptors for English persons though. Try calling them Sassenachs. 
…or “Sonnets from the Portuguese”?
“The English,” “the French,” “the Chinese,” etc. work as aggregate nouns. I don’t suppose there’s any logical reason why you couldn’t refer to an individual as “an English” or “a French” the way you would to “a Chinese”; but I’ve never heard it done that way.
There is an inner category for which no such noun exists (England, Ireland, Wales, France, Cornwall, Netherlands). Then there is a middle category where there is a special word that is different from the adjective (Spaniard, Scot, Swede, Turk, Jew, Dane, Pole, Arab). Next you have lots of nationalities where the adjective doubles as a noun (Italian, Norwegian, Russian, German, Austrian, Greek, Pakistani). And then you have nationalities where the adjective ends in “-ese”, which sound problematic when used as a noun (Japan, China, Vietnam, Portugal).
Kiwi.
New Zealander for when that needs to be explained.
Or rather, the gender-neutral form also serves as the masculine form.
Not the ones in Ticino. You’d be surprised at how many Americans are unaware that there is an Italian-speaking part of Switzerland. Or maybe you wouldn’t, considering that most of them probably a. couldn’t tell you exactly where Switzerland is, and b. couldn’t tell you anything about it other than that watches are made there, the Swiss Army Knife exists, there is a type of cheese that comes from there, and there are mountains. Maybe they’d know about Swiss banks if they’ve watched a lot of movies.
“Portuguese” sounds plural in English. But “Portuguoose” is… just wrong. 
I haven’t ever heard anyone referred to in the way that you describe.
And yet, through the magic of the Internet, it’s easy to find examples of people using those terms in that way. Just Google for phrases like “a Chinese from” and you’ll turn up plenty of relevant web pages. Or if online sources aren’t a reliable enough indicator of educated language for you, Google Books finds lots of dead-tree literature, including books written by American academics and published by university presses.
I find the terms odd-sounding, and I probably wouldn’t use them myself, but I can’t deny that they’re in use by others, and that these uses have made it past reputable editors of modern English literature.
Bangladeshi is a noun too.
In English it’s easy to use a noun as an adjective. It’s not as easy to use an adjective as a noun. So we’re still working on it for some countries
There’s some recent tendency in English to avoid singular demonyns that are also adjectives as somehow impolite, amidst a generally raised consciousness of formerly correct but now (claimed to be) impolite words especially related to nationality, race, ethnicity, etc. But it’s actually a little hard to believe you’ve never heard (it would be harder still to believe you’ve never read if you read much) a person from Japan or China referred to as a Japanese or a Chinese. Similarly, if perhaps a little different in ‘feel’ somehow, I’ve seldom heard a person from Korea described as other than ‘a Korean’, which is what Koreans are taught is correct English, and only a small % of Koreans who have both mastered English then also been exposed to and have adopted US-style PC speech would insist it has to be ‘a Korean person’.
In the Chinese, Korean and Japanese languages nearly every descriptor of people is followed by a Chinese character for person (often 人=ren/in/jin respectively but sometimes other characters meaning person) or the indigenous word ‘saram’ in Korean, but in English up to now that’s not generally necessary, even when the a demonym and adjective are the same.
The problem with using ‘a Korean’, or ‘a British’ or ‘a French’ is that you expect a noun to follow(e.g. a Korean writing system, a British accent, or a French military victory). So we get expressions like “a room full of Brits, Kiwis and Aussies.
Well, they call themselves Nederlander
Suppose you could also call them Dutch