even then calling it an organization is not really that accurate

It is an organization, you can go work for the NATO directly. They are headquartered in Brussels.

That said, military intervention on behalf of NATO works as you described, but there would be an obligation to help Canada in your example if the war would spread to its home soil. That said, the help obligation is literally worded “as they deem necessary”, so they could pretend that no large-scale intervention is necessary.