We are unlikely to become a republic in Canada but that does not mean we could not elect our defacto head of state.
In Canada our de jure head of state is the monarch of the United Kingdom who is then represented by a viceroy federally and provincially. Yes, our provinces have a head of state even though they are not independent countries. A problem in Canada, and other commonwealth countries other than the UK, have viceroys that are chosen by the prime ministers of the countries.
In the UK the prime minister does not choose the monarch. David Cameron meets with the same Queen that Churchill met with 60 years ago. Cameron is the 12th prime minister to serve Queen Elizabeth. It is an important distinction that in the UK the prime minister can not appoint the monarch but that in countries like Australia, Canada, Jamaica and New Zealand.
In the past the viceroys where chosen by the monarch from people in the UK and quite rightly the countries on the receiving end sought to change this. Canada led the change with first local being appointed as viceroy. This is the list of when each country first had a locally chosen person as their governor general:
1952 Canada - Vincent Massey
1965 Australia - Richard Cassey
1967 New Zealand - Arthur Porritt (Bernard Freyberg was Kiwi connected and governor general from 1946 to 1952)
1973 Jamaica - Clifford Campbell
The term of viceroys is not decades, it tends to be four to five years. This means they are only likely to be around for one or two prime ministers and only a single election. The term is too short to allow for a viceroy to rule in the name of the Queen for much beyond the term of the prime minster that chose them.
There is no reason the person recommended to the Queen as a viceroy could not be someone the people chose in an election. There is also no set term for how long the viceroy should be in power.
What I would suggest is that the governor general of Canada and all the lieutenant governors in the provinces be elected for a ten year term in a non-partisan election. I would let them run for a second term if they would like. All it would take would be a Prime Minister willing to do this and small amount of legislation in parliament.
The ten year term of a viceroy would mean that the viceroy should last through two to three first ministers. Allowing for more than one term could allow for viceroys that are around for a long time and given the Crown a stability and gravitas it is easily lacking in Canada at the moment.
A non-partisan election for the position would take any connection for formal political parties out of the mix. It can be done and is done in other countries around the world that elect presidents who are figureheads.
This change would make a small but important reduction in the power that sits within the Prime Minister's Office.