The thought of applying directly to the McKinsey New York office from Sydney never crossed my mind. Is this possible?

The broader question here is the feasibility of applying to international consulting offices. Short answer is possible but tough. Long answer is it comes down to a variety of factors - such as if you're a good fit for that region (language expertise? local work experience?); the quality of their local recruiting class. Typically if you list international offices on your application and are a strong consulting candidate, they'll consider you for the office, provided language is not a barrier.