It's been discussed before on this forum that HRH Prince Charles put himself forward as a suitable Governor General for Australia and was rebuffed. It's been common knowledge in Australia for decades, and although most were unhappy at the personal hurt that it may have caused him, most were accepting that this was the right decision. Personally, I have my doubts, despite my belief that our Governor General should be an Australian, because there's been only one, in the last few decades, whom I, as an Australian, have been proud of, viz., Sir William Deane.
Prince William, through no fault of his own, wouldn't be acceptable. If his father were thought unsuitable, there's no way that young William would ever be.
And from my limited observation of Canadian politics, I agree that their current Governor General is an inspiration and a wonderful representative of Her Majesty and the Canadian population, both.