As far as Spain is concerned, the new monarch still needs to be officially recognized by Parliament. Sure, it is merely a formality, but even so there is always a chance (however slight) that events will not go as expected.
And, since the monarch (King, in this case), reigns, but doesn't rule, yes, we still need a president (not, of course, in the Republican sense) for the Government.
Now, if what we Spaniards call the President is referred to as Prime Minister in English, that would disentangle this debate a bit, of course.