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If the President will be so useless that it's alright for the government to decide who it should be, we may as well have the Queen. If the President has an important role, it's the voters and not the government who should choose.
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It doesn't have to be either/or.
I think the system in the Republic of Ireland is quite good, in that the President is mostly Head of State with little executive power - she can delay a dodgy looking law by referring it to the Supreme Court and she can refuse to dissolve the Dáil (Parliament) if she reckons another government can be formed out of the same one before its term is up - and not much more than that. Not falling asleep at State dinners is the other main duty. But yet she still has to be elected by the people. Usually.
I say 'usually' because it's quite difficult to get nominated. You need the backing of 20 parliamentarians, from either or both houses, or 4 county councils.
When Mary the Second was elected, we had 5 candidates - including a man! Three were party political, and two were independents with county council backing. It made for a good contest.
And even though Mary only has similar powers to Elizabeth, she is in a much better position to be able to speak her mind in public - e.g., recently about the catholic church & child abuse - and, if she wishes, to give the finger to the government of the day because she has been directly elected by the people (and not by parliament, like the last Australian proposal). Not even the Taoiseach (Prime Minister) has that moral mandate.
Term of office is 7 years. You can go for a second term but no more more than that. Mary II got in unopposed for her second term - and, yes, there's questions about the fairness of how difficult it can be to get a nomination to stand against a sitting President.
Still, the option of a popularly-elected President but with mostly ceremonial powers is one that I would recommend to Australian republicans.