The republic of Tim Horton... ...and Timbits for all.
I think the constitutional changes are the biggest hurdle we would face... and it's a damn big one. There would have to be a very big personality (on the level of a Trudeau) to push that kind of change through...
You are correct that the current role of the GG is essentially apolitical. If it became an elected position, not only would it cost *more* money to maintain the position but it would become pointlessly partisan.
I don't see that multiculturalism plays into this... in fact, the last two GGs have been immigrants (Clarkson from China and Jean from Haiti) which in and of itself is interesting. While I can agree that our adherence to the Monarchy is kind of charming, if more than a little outdated, I don't see that it is a negative as far as the multicultural make up of our nation. Hell, the UK is as multi-ethnic as we are and they have the actual Queen rather than a rep.
In the end, it is a symbol of our past. The key, and I think Clarkson and some of the more recent GGs like Hnatyshyn are on the right path, is to make the position more relevant to Canadians. More visible both here and abroad. It may cost a little more money but in the grand scheme of things I'd rather see it spent on this sort of thing than programs (legit or not) like the Sponsorship money. It seems to me the GG is in a great postion to promote the idea of Canada and federalism to Canadians (apoloitically speaking).