The first Canadians to win a anything were the "Paris crew" who won a regatta days after the constitution. Canada started to be represented in the Olympic Games in 1900.
https://www.thecanadianencyclopedia.ca/en/article/paris-crew
https://en.wikipedia.org/wiki/Canada_at_the_1900_Summer_Olym...
If Canada counts as not the Crown, then so does NZ.
It's a major oversight for an article on this subject.
https://en.wikipedia.org/wiki/Statute_of_Westminster_1931
>>Most of the remaining colonies in North America – everything north of the United States with the exception of Newfoundland – were merged into a federal polity known as "Canada" in the late 1860s and early 1870s. Canada was termed a "dominion", a term previously used in slightly different contexts in English history, and granted a broad array of powers between the federal government and the provincial governments. Australia was similarly deemed a dominion when it federated in 1901, as were Newfoundland, New Zealand, South Africa, and the Irish Free State in the first decades of the 20th century.
>> No Act of Parliament of the United Kingdom passed after the commencement of this Act shall extend, or be deemed to extend, to a Dominion as part of the law of that Dominion, unless it is expressly declared in that Act that that Dominion has requested, and consented to, the enactment thereof.
In that language was/is Canadian independence from britain.
Unlike, say the US, independence for the former Dominions isn't something you can pin point to one event. It is mostly a slow evolution of the legal environment and cultural separation over the course of more than a century. (In NZ's case you could argue that it took 163 years, or is not done yet because reasons.)
The actual difference between the Canadian "expedition" and the one that included Hillary is that the Canadian one was just some dude. One guy that wanted to try his luck, whereas the British one was organised (and I believe paid for) in Britain. It just happened to have a Kiwi on it.