I don't see any compelling need to make any changes to the G-G as an institution, or the Senate for that matter. Apart from all the good constitutional information provided by our resident expert FiveParadox (that name always makes me think of 10 mallards flying over a Ukrainian settlement in Saskatchewan... ), the G-G has another important role: the pomp and ceremony associated with being the Head of State. The U.S. President, for instance, is the nation's Chief Executive Officer as well as being Head of State, so there are a lot of ceremonial duties he has to perform when he could certainly be doing something more important and useful, like his executive functions. The Prime Minister in constitutional monarchies like Britain, Canada and Australia is the CEO, but not the Head of State, that falls on the monarch or the G-G, so the tiresome business of extending formal courtesies to visiting Heads of State from elsewhere, throwing parties for foreign dignitaries, cutting ribbons at supermarket openings, etc., is not a necessary part of the PM's duties.
If we do away with the G-G, somebody will still have to perform those functions the G-G currently does, those duties won't go away. I suppose we could do something similar to the French model, with both a Prime Minister and a President, but except for electing a President, we could get pretty much the same effect by simply changing the title from Governor-General to President. And if we start electing people like that rather than appointing them, suddenly they have a political legitimacy they've never had before, and there's no way to predict what effect that'll have. It's the same argument against electing the Senate: it'd give the Senate a political legitimacy it doesn't currently have, and the consequences are not really predictable. Abolishing the G-G and electing the Senate may seem like simple and obvious things to do, but they're not, they have deep and complex implications that go to the heart of our constitutional arrangements.
Bottom line for me: nothing's broken in this context, so don't try to fix it.
edited to fix format errors. I hate having to admit I make mistakes...