To add to this, Canada in 2008 went through a "parliamentary dispute" that Canada's Governor General played a large part in the resolution of. While the Governor General is pretty much chosen by Canadian ministers in a democratic fashion, on paper the officer is appointed by the Queen and ideally represents her in Canada.
So I do agree that the monarchy has some influence on the Commonwealth, more than just having "the right to be consulted, to encourage and to warn".
So I do agree that the monarchy has some influence on the Commonwealth, more than just having "the right to be consulted, to encourage and to warn".