There’s also the problem of what to replace it with. For example, in the United Kingdom, and in monarchies modelled on the U.K., such as Canada and Australia, the monarch has, on paper, a lot of legal authority - the power to dissolve Parliament and call elections, to appoint and dismiss the Prime Minister, to refuse assent to a bill, and so on. Of course, the Queen doesn’t use those powers on her own bat, but only as advised by her Prime Minister. But it’s not because of legal restrictions on her ability to do so, but because she has no politicial legitimacy to use those powers against the advice of the democratically elected leader of the Government.
But if you just substititute a democratically elected President for the Queen, suddenly you have someone who has political legitimacy and therefore might be willing to use those powers, drastically changing the balance of power between Parliament, the Prime Minister and Cabinet, and the Head of State. (See the French system for an example.) That’s fine if you want to make that substantial a change, but the parliamentary system seems to be working well, so why would we want to suddenly have a powerful head of state, breaking with centuries of parliamentary tradition and Cabinet government, responsible to Parliament?
So if you want an elected President, but also want to preserve the current system of Cabinet government responsible to the Commons for all major government decisions, you have to make substantial constitutional changes to the powers of the Head of State.
Alternatively, you might want to consider a form of indirect election of the Head of State in the new republican government. If the Head of State isn’t popularly elected, he or she will have the same lack of political legitimacy as the Queen, and therefore won’t be able to use the powers without the advice of the Prime Minister. But is indirect election, in and of itself, enough to keep the Head of State in a figurehead role? Or do you still need to make major constitutional changes to the powers of the Head of State, maybe even defining the office of the Prime Minister and his powers? But that would be a major change as well - currently, the Prime Minister has no powers defined by the Constitituion, and only has the authority granted by statutes passed by Parliament, and by the well-defined constitutional conventions of responsible government. Defining the Prime Minister’s powers in the Constitution would be a major change.
And then, what form of indirect election is sufficient to satisfy republican instincts, but sufficiently indirect that it doesn’t confer polticial legitimacy on the Head of State, other than in a caretaker function? That can be politically difficult, especially in a federation such as Canada or Australia, where there also has to be regional balance. Even if you do come up with an indirect election, it still may require significant consequential amendments to the Constitution, relating to the powers of the new Head of State.
In short, it’s not a simple change to make, in terms of constitutional and political balances. If you have constitutions that are premised on a monarchy, that permeates the system in a lot of ways - and anyone trying to change that has to think about all of those implications.
All of this has the potential to use up a lot of political time and energy. If the system is humming along pretty well, where’s the need?
One way to think about these issues of constitutional design is to flip it around - suppose there was a desire to go to a parliamentary model in the U.S. - would it be a simple thing to leave the President with all the powers of the office, and yet implement constitutional restrictions on the use of those powers so that the real power lay with the leader of the majority in the House of Representatives? Or would major constitutional amendments be necessary?
Some of our Australian Dopers can no doubt comment on this issue in more detail, since they went through a referendum a few years ago on keeping the queen. My understanding is that even though there was considerable sentiment in favour of going republican, the proposal failed because of the difficulties in designing a new Head of State position.