FFS. This is an extremely tedious issue that has been done to death.
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Look obviously I respect the Cooks, I respect everyone developing the game, but the fact is that fundamentally they should not be allowed to compete in the World Cup, and trying to defend their inclusion using BS excuses about the Home Nations etc is the exact sort of thing that has made our sport such a pathetic joke at international level for so many years.
This issue was brought up, merely because generalised assumptions are made about what a nation, state, country, sovereign state are, without considering different definitions of the term.
People from the Cook Islands are all technically also from New Zealand, because Cook Islands is part of New Zealand. Therefore, all Cook Island players qualify for New Zealand, and therefore when Cook Islands and New Zealand play in the same tournament we end up with Cook Islanders having two teams to represent them, which obviously is wrong.
The Cook Islands is a self-governing parliamentary democracy since 1965, before that time it had been annexed in 1901 by NZ. Cook Islands is an associated state. What that specifically means is that NZ has responsibility for CI's defence and foreign affairs, which seems logical given that CI's population is less than 20,000, and NZ citizenship means CI people can enjoy the freedom to live/work in NZ.
This point of the Cook Islands having two teams to represent it is incorrect, but even if it were correct, it would be one example of a trend that happens in a range of sports, as shown below.
The Home Nations are not territories of any country, people born in the Home Nations do not qualify to play for more than one national team simply by birth. Therefore your example is not relevant at all.
Actually, people born in Northern Ireland qualify by birth for both the Northern Ireland soccer team, and the Republic of Ireland soccer team, due to the Good Friday agreement, and the fact that under the ROI constitution, people born in NI are automatically entitled to ROI citizenship. The ROI/NI soccer eligibility ruling has been validated by the Court of Arbitration for Sport (CAS). Therefore, Northern Ireland is effectively represented by two teams in soccer (NI and ROI), and in the Olympics (both the UK, and ROI teams). The Irish Olympic team claimed for years to represent NI, and with the citizenship ruling, can do so.
New Caledonia is an overseas department of France, a special collectivity, New Caledonian Christian Karembeu qualified for France's soccer team by being from a New Caledonia, an overseas territory of France. NC was not a member of FIFA then, but it is now, and in theory, NC could make a World Cup and play France, and that territory would have been represented by two teams. There are a host of territories in FIFA represented by two Associations.
Netherlands Antilles took part in the Baseball World Cup in 2009 alongside the Netherlands. The Antilles and Aruba could take part in the next World Baseball Classic in 2013. Both could represented by the territory team and by the Netherlands team.
Puerto Rico is an unincorporated territory of the USA, its people are US citizens, but it has an Olympic team, and it has played against the USA in international competition in basketball and baseball. Indeed, Puerto Rico defeated the USA at the Athens Games in 2004.
Why Puerto Rico Has Its Own Team
American Samoa is an unincorporated, unorganised territory of the USA, its people are US nationals, and can travel and reside freely in the USA. They play in FIFA competition, and could play in a World Cup theoretically if they qualified, and thy do send a team to the Olympics. Thus, the territory could be represented by two teams.
The nation, state or region (depending on your POV) of Taiwan/Chinese Taipei/ROC is arguably represented by two teams i.e. Chinese Taipei, and the People's Republic of China team.
Different sports have different approaches in defining, recognising and including so-called "national" teams, just as different societies have different ideas of what a "nation" is.