A major weakness of the American system of democracy is the office of the president itself. By combining the role of head of state, head of government, and party leader into a single office the American Founding Fathers created a position which is unable to do any of its assigned roles particularity well. As head of state the president is forever in conflict with himself over doing and saying what is good for the country as opposed to what is good for his party. As head of government the president's focus is divided between keeping his party united and his country working in harmony when he should be focused on crafting effective legislation. As party leader the president is too far removed from the party itself (and too busy) to effectively guide its actions. As a result the president's every action is viewed with a cynical eye, every bill before congress a disorganized and bloated mess, and both parties lacking in clear and transparent structure. And pity the poor fool who becomes president as the four to eight years drain the life from them and leaves them visibly aged. The pressures of the office of president mean that only very specific types of people seek to become president and it is debatable that they, on the whole, have the characteristics for being a good head of state. Even before the last vote has been cast the successful candidate will have a scarred country to heal. A country scarred partially by their own efforts to attain power. And even if they are successful in healing the damage they have done they must contend with the partisan shots fired from the opposition ranks. If they accept these shots as legitimate (aimed as they are at the head of government and party leader) the office of the president becomes covered in muck. If on the other hand the president wraps himself in the flag, constitution, and republican virtue the needed scrutiny of legislation is diminished as the cowed opposition, unwilling to appear unpatriotic, acquiesce to whatever law their god-president demands.
Finally, it has been shown that directly-electing a president decreases voter turnout for legislative elections by 5-7% (among other things). 
The Unworkable Solution
Some countries have become parliamentary republics in order to separate the office of head of state from that of head of government and avoid potential pitfalls. They are only marginally successful. The president of such a system serves a ceremonial role and is either a powerless nobody or more inclined to interfere in the legislature's business than one would generally allow of a position appointed by said legislature.  How much interference? Enough that there is a measurable difference in how parliaments of constitutional monarchies, presidential republics, and legislative republics handle government failures. To be specific, both forms of republic have a marked preference for not consulting the people via election when governments fail.  But even if such things are avoided, who becomes president in a legislative republic? Politicians, thinking very highly of themselves, cannot comprehend of a more fitting person to be president than a fellow politician. So it is that even if a republic starts off naming accomplished members of society from outside the political sphere eventually it begins to favour politicians. 
The Royal Solution
A constitutional monarch suffers none of the disadvantages of a president. By being solely head of state they can carry out the role of being a national uniting figure and a non-partisan focus of loyalty. The people can be assured that when they speak, they speak of their own convictions without an overlay of partisan considerations. By retaining reserve powers the monarch acts as the final defence of the constitution, and democracy, against those who would harm either. By being irremovable from the pinnacle of power they remind politicians that they too are servants of the people. By wrapping themselves in patriotic glory the monarch hinders politicians from likewise wrapping themselves in the flag. By being a constant in a country's life for many years the monarch is a symbol of stability in a chaotic world; a comfort. And whats more, they can do it full time while the head of government works at their own duties. Perhaps even helping the populous to trust each other more now that there is no president tearing them apart.  And by leaving law-making up to Congress the political class can no longer point fingers at the executive when they fail to come to agreement. There may even be positive economic effects to having a monarch. 
And by putting a position gained by that one commonality we all share, the accident of birth, at the center of the nation Americans would be affirming their faith in human equality. For by doing so we triumph nature over political maneuver, the individual over the faceless state, and those doing their best over those especially adept at politics. It is the victory of the human person.