Could the GG easily become head of state instead of royalty, with the process of the government choosing the GG remaining as it is now?

Very easily, Royals are surplus to requirements. Government could easily pick their own head of state. its not rocket science.

If the Canadians decided to do away with the monarchy, they'd also have to decide how to choose their head of state. It wouldn't be automatic that the GG at the time would become the head of state. Instead, they would most likely hold elections. The GG would cease to be GG as soon as the monarchy was abolished but could then be a candidate for head of state.

Or, perhaps, they would just decide that whoever was head of government was also head of state, as we do in the US. Head of state in most countries where there are two figures is really just a ceremonial position. In Italy, half the Italians can't even remember who the head of state is.

The GG is appointed by the Crown. On the recommendation of the government, sure, but, the power to do so still rests with the Crown. Having Parliament do it would be like selecting your own boss. Possible, but, there would be a lot of political fighting. Technically, it could be done, but, it would not be easy at all.

The GG is the monarch's representative in the country in question.

Until WW2, Canadian GG's were usually British citizens, usually from the nobility. Due to changes such as the Statute of Westminster and Canada's growth during that war, it was more than clear that such representation should become more Canadian, so that now, the GG is a Canadian, appointed by the Prime Minister, and the British monarchy agrees to the Canadian PM's pick.