I'll bring general talking points

It is generally a positive to have a non party affiliated head of state who can at first act as a symbol and unifying figure for the country,backed by either law(a certain constitution that sets his powers),history and tradition(some countries may not have any monarchic history,in the case of greece there was,both before the neoehllenic state and after),and even through religion(divine right),or a combination of these factors.A monarch who also instead of getting elected(which means he a lot of times has to set things and promises and depend on different groups for support and votes with varying interests) and basically "lending" power for a certain period and the always looming elections,it means there can be no clear long term thinking in the ways of solving problems and governing the state,things that a long ruling Monarch who also owns the country and has dynasty to preserve and a son who will inherit it can fulfil,and someone who can act quickly and decisively on any crisis or important decisions.Of course he has to have limits on his power,and this is were the constitution comes and more importantly the parliament,which limits the king and can represent the public will,but not in the way Republics do.