Most ethnic/national groups have a gender neutral noun to refer to them in English: a Pole, an Italian, a Spaniard, etc.
But there isn’t a gender-neutral noun for some groups, like France, England, Wales or Ireland. The terms are a “Frenchman”, an “Englishwoman” and so on.
How come?
For England, Scotland, Wales, and Northern Ireland, what’s wrong with ‘Briton’?
There is no requirement that language be consistent.
Here’s a completely just made-up theory that may bear little to no relation to the truth. English was invented in England, so they would be more likely to refer to someone by their city or region rather than the conglomeration as a whole. Similarly, France as an entity was constantly shifting and in very regular contact with Britain when English began to solidify in the Early Modern period, so they would be more likely to be referred to as Normans or Burgundians than ‘Frenchmen’ as a whole unit. Discuss and critique as you will.
Same for Ireland I believe
Because I have Welsh and Scots relatives who reject the term “Briton.”
Plus, there are times when you want to refer specifically to someone from one of those regions and “Briton” isn’t specific enough.
Other nations for which there isn’t a non-gendered noun for an individual: Bangladesh, Burma, China, Japan, Lebanon, Netherlands, New Zealand, Portugal, Switzerland. This will likely become an issue of dialect.
What noun do you use to refer to someone form New Zealand other than the neuter “New Zealander”?
I can’t think of any noun for someone from Switzerland other noun phrases. I’ve never seen Swissman but only Swiss man. In any case, surely Switzerlander would be understood. And while those from the Netherlands are commonly called Dutchman or woman, a “Netherlander” seems fine.
You have “Englisher”, as well as “Angle”, “Englishperson”. If the suffix “-man” is too sexist in context, it is easy to avoid using it: “The French are European”, “She is English, not Russian”.
“Limey” isn’t gender-neutral?
ducks and runs
I’d say a “person from New Zealand”. “New Zealander” sounds as awkward as “Netherlander”; understandable but strange.
Also “Frog”.
joins the evasive maneuver
Bangladesh - Bangladeshi
Burma - Burman
China - ?
Japan - ?
Lebanon - I usually use their ethnicity i.e. Druze, Marionite, etc.
Netherlands - Netherlander
New Zealand - New Zealander
Portugal - ?
Switzerland - A Swiss or a Switzerlander
Seems to me, and this is taken from my left elbow, as if words such as Englishman are simply the union of the longer adjective+(wo)man forms which exist for other peoples. Whether the majority of peoples get the adjective+(wo)man version, the adjective(wo)man version, or a word which is both noun and adjective requires more time spent counting than I can be bothered to spend. It does seem to me as if those where there are a noun and adjective which are different and which did not originate as slurs are the definite minority.
“He is a proud New Zealander” sounds fine to me. Granted, I don’t discuss New Zealand very often.
Lusitanian 
From an Aussie, New Zealander is fine and used regularly in Oz and NZ. Strangely though I agree that Netherlander sounds strange to me as well.
Is this some sort of Whoosh with a capital “W?”
While I have heard of “Chinaman” and “Dutchman” I’ve never heard of “Japanman” or other gendered variations of most of these counties.
I think the idea is that for those countries, there isn’t a singular noun at all, gendered or non-gendered. You’ve never heard of “Japanman” or “Portugalman” , but we also don’t (usually) refer to an individual simply as a “Japanese” or a “Portuguese” using the adjective form as a noun, as happens with “Italian” or “German”.
What you are after is called a “demonym”, and people do compile lists of them:
Perhaps the awkwardness of “New Zealander” is why “Kiwi” became commonly accepted and not viewed as derogatory.