Not according to the CIA Factbook it isn't:
NZ: European 71.2%, Maori 14.1%, Asian 11.3%, Pacific peoples 7.6%, Middle Eastern, Latin American, African 1.1%, other 1.6%, not stated or unidentified 5.4%
USA: white 72.4%, black 12.6%, Asian 4.8%, Amerindian and Alaska native 0.9%, native Hawaiian and other Pacific islander 0.2%, other 6.2%, two or more races 2.9% (2010 estimate)
I would say almost quite comparable.
There was a recent article where a citizen of a Nordic country got upset at this common view, pointing out that perhaps it seems homogeneous to the US because in the US, today, you are white, black, hispanic, asian etc. There are not strong subdivisions in these categories, it seems, and subsequently they apply this view to other countries. But in Europe and Norway, Sweden etc, there are many flavours of 'white' with a long and recent history of animosity.
NZ is a country with a large indigenous population (the Maori), a lot of pacific island immigration, and increasing and accelerating immigrant population of Indians and Chinese (among others). These all assert social pressures of their own, and I would argue its partially our social safety net and maybe certain aspects of our culture and economic structure that have allowed it to be (mostly) harmonious, so far.
Not saying your entirely wrong, but that it is not (no pun intended) black and white.
Being a democracy, the policies of the country are inseparable from its culture and demographics. You say universal healthcare might stem the tide of cultural breakdown. But we cannot be sure the causality doesn't flow the other way. Unfortunately, it's likely a feedback loop. We have bad policy in part due to the cultural and social decline the author describes, and we have cultural and social decline in part due to bad policy. Simply proposing policy changes neglects the complex interdependencies of these things, and is why national demographics are so important in a democracy.
Part of the reason NZ is able to cohere around its policy more than the US is because everyone in NZ (to a greater degree than in the US) shares the same values and cultural identity. This is why you see similarly idyllic societies in the Scandinavian countries. High social cohesion due to shared cultural and ethnic identity. Same in Korea and Japan - although Japanese culture is a good example of some of the downsides of ethnic and cultural homogeneity as well.