New Zealand and Australia were part of the British Empire and are still members of the British Commonwealth, as is Canada, while America became independent in 1776. As a result, there was greater attraction for British settlers to nations under the Union Jack and less of one for non-British Europeans. Additionally, both the U.S. and Canada are closer to Europe than are Australia and New Zealand. With respect to Dutch and German preference for America as opposed to the British Empire/Commonwealth nations, you must remember that there was a strong Dutch and German presence in colonial America, concentrated in the Middle Atlantic colonies of New York, New Jersey, and Pennsylvania. Even today, the concentration of German Americans in Pennsylvania is one of the highest in the nation.
There was also a lot of discrimination on non-British Europeans on the part of New Zealand. There were a number of documents in the government archives at Wellington's National Library that showed post-WWII New Zealand governments discouraged Italians, Greeks, and German immigrants because culturally "they are imcompatible with the values and life of a majority of New Zealanders". Dutch and Scandinavians were reluctantly admitted since there were fairly similar to British.
Policies were in place to favour British/Irish migrants right up to the 1970s. Until 1977 any British and Irish national landing in New Zealand could register and become a NZ citizen. At times the New Zealand government even paid for the entire cost of voyage from Britain or Ireland: they were called "10-pound migrants". It has always been said that New Zealand is a kin migrant country, and that it is not a New World country but rather, a New Britain.