As I mentioned, the GST is administered by the Government of Quebec for the federal government. The Quebec government is paid a contribution for the work it does. Apart from that, I cannot really answer your question. As I told you, I am not going to make something up. The situation is historic. I started working at the department in 1996, when it was called Revenue Canada. From my perspective, the distribution has always been as it is now.
On May 6th, 2008. See this statement in context.