It seems odd today, but back in the 19th century as many nations were being transformed into modern states, Constitutional Monarchy was certainly one of the more popular forms of government. It seems quaint and archaic because few today see a role for a monarch. And yet thinkers of the time understood that it was important even in a parliamentary or constitutional system to have someone who could function as caretaker.
In several countries, the monarch was replaced by the president and thus it's not unusual to find a prime minister as head of government and a president (and sometimes a king) as head of state.
There are many variations and arrangements in how this can work. Sometimes the president or king as head of state is a merely ceremonial role in other cases the head of state will have specific (if limited) powers.
The point is, in numerous countries you have a figure that can step in and intervene when things get, shall we say, out of hand. Iohannis has intervened to block the appointment of a prime minister. In Italy with the resignation of Prime Minister Renzi the president steps in and starts to re-organise the government. He acts as a temporary leader or custodian. In Spain with the end of the Franco regime it was the monarch Juan Carlos who stepped in and helped shepherd the country toward parliamentary democracy. Even recently during several failed attempts to form the government the king, in this case Felipe VI, steps up and once again fulfills that 'shepherd' role and then when the government is formed, steps back. It's also noteworthy that in 1981 Juan Carlos intervened and played an important role in blocking a coup that sought to re-establish the Right-wing military dictatorship.
Sometimes like in Romania the actions of the head of state risk fomenting a constitutional crisis. We'll have to wait and see.
In the United States the roles of head of government and state were combined in the person of the president. He has powers that go beyond those of a prime minister and then in addition as head of state fulfills a ceremonial function and takes on something of that role as national shepherd even though the US Constitution allows for no 'transitional' scenario. Because the US is not a parliamentary system, No-Confidence votes and Snap Elections are not a possibility. The government cannot resign.
The one institution that's supposed to (somewhat) serve in this capacity is the Electoral College. Supposedly if a completely unviable candidate was selected the college could reverse the popular vote. And yet apart from a few individual reversals the very fluid, unique and temporary institution has never dared to do so. In fact if they did, it would provoke a constitutional crisis.
The perfect opportunity was in 2016 with Trump's victory in the presidential election. He lost the popular vote by a significant margin and certainly was one of the most unpopular and unqualified candidates to ever win the election. In terms of the Electoral College, he won but once again when the college met in December to actually decide the presidential race there was (hypothetically) no guarantee or even legal requirement that they follow the popular vote. The usual winner-take-all method of the college strikes many as rather lame and in the case of 2000 and 2016 meant the candidate who lost the popular vote, but won the right combination of states won the election.
If we had a system in which the head of state was a figure apart from the president, we could have had an instance like what just happened in Romania. The head of state could step up and say, 'wait a minute, this won't do'. And he could have potentially forced a new election.
In some ways this system is understandable and provides an interesting safeguard.
Of course if the Electoral College had reversed the election, you would have a significant bloc within congress that would have balked and there would be a strong potential for civil unrest. Apart from the one episode of civil war, the United States has been able to force through sometimes controversial transitions of power. And yet if elections continue to be controversial as they were in 2000, 2004 and now 2016, the years of smooth sailing may come to an end and the American system has no real mechanism of intervention and/or temporary custodianship.
At that point some might wish for a king, not as a ruling monarch but as one who can intervene. Even though Britain's monarchs no longer exercise power, they still can quietly exert some influence and the outward forms of monarchical ratification are retained. The Queen formally asks incoming prime minister's to form a government. Sitting prime minister's go the monarch to tender their resignation. Elizabeth goes to the parliament each year and presides over its opening. Though the monarch doesn't exercise real day-to-day power and Charles recently upset many when it was revealed he was receiving intelligence briefings, nevertheless the monarch waits in the wings as it were and exerts some influence. If trouble were to arise, the monarch could (as in Spain) step in and provide leadership during a time of transition.
There's something to be said for such a system and for all the clever mechanisms of balance found within the US system, it is a far cry from perfect and has in many ways painted itself into a corner, a reality that becomes clear during times of crisis.
An interesting additional note with regard to Romania:
The king in Romania played a much more forceful role during World War II. Michael I played a significant part in the coup that removed pro-Nazi dictator Ion Antonescu in 1944 and moved Romania from the Axis into the Allied camp. When the communists took over in 1947, Michael was forced to abdicate and go into exile. The monarchy was subsequently abolished.
He was allowed to permanently return to Romania in 1997 with his citizenship restored. There was a real possibility of the monarchy being restored in the 1990s, so much so that his visits in the early parts of the decade were controversial. But now as the decades have passed it's become clear the will be no restoration. The president is head of state.
I mentioned him for two reasons. One it's an extreme example of the head of state intervening on behalf of the nation and two... many people have forgotten as of December 2016, he's still alive. He's the last of the important World War II leaders. Because the Eastern Front is almost ignored in the West, he has received very little attention and yet I find it fascinating that this man, who was installed as a puppet of the fascists, had met with Hitler and Mussolini and yet dramatically rose up and helped throw down his own nation's Right-wing dictatorship, still walks among us.
There are many World War II survivors still alive, although they're all very old at this point. Michael is the last of the public figures, a leader who played a not insignificant part in the course of the war. A living relic, he'll be gone soon. He's 95 and in poor health.