seems to me the problem with the republic thing is everyone gets to vote for head of state, so when you get some lunatic orange con-man or some dizzy old geezer in it looks bad on your country.
Whereas with a monarch its hereditary so no-one is to blame if they are rubbish/embarrassing/do a flit to Argentina with all the bunce.
One way round that would be to have being head of state as a lottery prize - 'It could be you!'. And you get to do it for six months or maybe a year then it's someone else's turn.
That way we don't get the moaning about 'unearned privilege' and it's no-one's fault if they're crap.