Whatever the advantages of monarchies, if any, were enjoyed by a selected number of few citizens and siblings [not always]. The majority suffered from heavy taxation and hard labour in the monarch's and noble's lands. It caused several revolts and wars over the powers of a monarch and the rights of civilians. The American Revolution was the first against the British rule, followed by the French Revolution against the House of Bourbons both in the 18th century and the last against the House of Romanov in Russia in the 20th century.
************ The above poster is actually deriving his understanding of Monarchy from popular Mythology, especially of the American Revolution. You can tell by the reference to High Taxation. The Truth is, most Monarchies in History did not actually place a high tax rate on their Subjects, nor were most who lived in those Monarchies forced to work Noble Lands. Heck, the American Colonies were free, with the people doing whatever work they liked, and they had the right to own land. The Taxes we hear so much about were the Lowest in the Empire, or the Civilised World.
Which brings us to one advantage of Monarchy, Decentralisation is possible and much more likely in a Monarchy. Republics tend to concentrate Power on the Central Government, which in turn builds huge Bureaus for the regulation of Society. This is done to ensure that the agenda of the ruling party is passed, and to help manipulate society to maintain the voters needed to maintain Power.
A republic is also run by those with Ambition to lead. Who else runs for Office. The Politician wants to have Power. They gain this power by being Charismatic enough to secure Votes. They also must belong to a group or party that helps them, and to advance in that Party must push string that party's Agenda.
So you get Ideologues pushing a radical Agenda.
Worse, there are other factions in society, who vote for other Parties. They will be demonised.
In the end, society is divided, as people turn on each other over Political differences, created by these Political party's pushing their agenda and making deals with special interest groups. The Politician to maintain power must also endlessly cater to the whims and desires of the Party and its special interest group supporters.
In contrast, the Monarch doesn't need to make deals with special interest groups and is not beholden to a Political Party. He is free to make his own mind up without fear of voter reprisal, or punishment from Political Party officials or special interests.
Likewise, there Is Stability in a Monarchy that cannot exist in a republic. The Monarch remains usually for life, and passes the Throne on to his successor, usually but not always his own Child. In this way the Uncertainty of elections is avoided, as is the division that creates social tensions and tear nations apart. Likewise, the Monarch has the advantage of Loyalty. In a Monarchy, one can be Loyal to the King whether or not one agrees with the Policies the King endorses, because the King rules by Right, not by Popular Opinion. Therefore, there can exit a Loyal opposition. In an elected Republic, the President rules by Popular Mandate, and against power by a Political Platform. If you disagree with that Platform you are effectively saying the president should not be President, and therefore are not really Loyal to him. There is no real Loyal Opposition. Look at the last two Presidents of the United States, Presidents Bush and Obama, and tell me how this is not so? They are or were rejected both of them by about half the Nation. Likewise, neither of them really United America, and by nature they can't. The idea of a Uniting Republic is simply a contradiction in terms.
Of course one can object and say that in a Monarchy we are still not Free, but such an objection assumes that our Freedom rests on our participation in elections, as I somehow voting for Politicians to hold power ensures they wont pass laws that constrain us, and an unselected Governor will somehow always remove our Rights.
Historically this has been untrue. In fact, the reverse has been True. Republics tend to pass far more Laws than do Monarchies. If you don't believe me look at today's world in which most monarchs are figureheads and Democracy reigns supreme. We have to get permission from the elected Government in order to make improvements on our own land, travel abroad, and n some cases work in certain jobs. The Government tells us what we can and cant use for barter. The Government regulates our food, demands compliance with certain Civil Rules, and regulates how our things are made. Monarchies in the Middle Ages, thought of by many as "The Dark Ages", never even tried to tell anyone how big they have to make a sink, how much fruit their allowed to sell for what price, or dictate how many hours a Store may be opened or closed. Republics do this all the time.
Monarchs tended to Respect Property rights, rather than overriding them for the sake of some community project. Monarchs didn't care if you sold your fruit for a dollar or two dollars. Monarchs didn't care how late your stores were opened. Monarchs allowed you to decide how you ran your own life and were simply not all that interested in Social Engineering. They, in short, tended to be more Free.
This was even true of France, were we are assured King Louis the 16th was a Tyrant, and the Revolution necessary. But the Revolution lead to the Reign of Terror and loads of people killed for not giving up Catholicism or being insufficiently Atheistic, not agreeing with the Revolution, or at the end not with whoever's faction in the Revolution was in charge.
The King had not even been a Tyrant, he was on his way to fixing the French Economic woes.
And that's another advantage of a Monarchy, the King can think Long Term. By Nature a Politician only thinks to his next Election. He has no personal stake in the Nation either. He is a mere Tenet. The King meanwhile owns the Government, and it is in his own interest to maintain it, for it is his. He also must look to his Posterity. It is the difference between a Renter and an Owner. Presidents are Tenets of the Capital Buildings they live in, Kings Owners.
So these are the advantages of Monarchy. They tend to be decentralised, for while the King rules he tends to let Local Matters be handled by Local Lords or the people who live there, they don't force Homogeny as to Republics, and allow divergence Cultures, it is possible to be Loyal to the King and not agree with his Policies, whilst a Politicians only claim to power is his Political Agenda, and to disagree with it is to not be Loyal to him, and Monarchs tend to allow Greater Freedom, and care for the Long Term good of the Nation. Monarchs unite the people, too, unlike a President is capable of.