From a storytelling point of view, monarchies can be a fantastic spark for conflict, even if the actual notion of governance is not deeply examined by the text, whether this is by committing acts of horror, starting a war, kidnapping a princess (or a prince for that matter!), declaring what modern readers would consider a positive change and the ripples reflecting out from that depending on the people's willingness to accept said change, dying and sparking a succession crisis, etc. The concept of absolute power can also have other characters strive towards it, sparking off other conflicts. And then there's the magical element. Cosmic agreements with elemental forces to protect one's kingdom are often easier to keep within a bloodline.
There are, of course, ways to explore monarchies beyond the traditional male primogeniture and Salic law. The system of tanistry depends on a monarchy where a king chooses his successors. The Holy Roman Empire had electors who could elect the new monarch.
I have several stories featuring monarchies, mostly because I enjoy the glamour (it's probably telling that my biggest writing projects are fantasy and Regency romances). However, monarchies can also be good for exploring concepts of abuses of power, the burdens of rule, and even the dangers of selfishness in rulers. In my biggest world, Middangard, the four stories that kicked off the world centre around the concept of Cadaln being in danger from darkness. This is human darkness, brought about by the Cyngor - essentially a high council - who have kept the king ignorant of the true state of his kingdom after a series of assassination attempts necessitated his going into hiding. As I've progressed with the world-building, this went from a few simple assassination attempts to a poisoning that killed the king's second child in the womb (by the mad monarch of another kingdom - he eventually helps start a war that his daughter ends up having to resolve and ultimately leads to his kingdom's absorption by another power, although his bloodline lives through his daughter's marriage to king of said other kingdom), assassination attempts on his son, and even the eventual murder of his queen in their 'safe' stronghold. While some threats are real, the power some members of the Cyngor acquired during King Harailt's seclusion led them to exaggerate the threat, ultimately keeping him and his increasingly frustrated son closeted away until the start of the quartet (specifically Crisiant's novel). While Crisiant's novel deals with the personal elements of this, both through their journey to the seat of power in Cadaln and Harailt and Connor's reactions to it, the other novels are designed to consider the further implications of it - one of the lines I want to echo through all four is "King or Cyngor? Make your choice." Thus the conflict of a government they cannot be certain is actually relaying their king's commands, versus a king some are saying has been bewitched and virtually abandoned them. Part of this stems from Harailt's original status as a second son - he did not grow up anticipating kingship, and therefore wasn't sufficiently trained up to it, and was possibly a little too trusting of his predecessors' Cyngors. This ultimately leads to Harailt's unhappy discovery of having failed his people, and his son Connor's need to fix those mistakes.
In another story, a very unfortunate king loses all his children, grandchildren, brothers, nieces and nephews, and is left with second-cousins as heir. Some other second-cousins decide to take advantage and strike at their competitors, and even some the next level down the line. This proves very stupid - they leave someone with a better claim alive, and the survivor of the third-cousin they attack is the thirty-six-year-old you-just-made-me-lord-and-a-widower. He leads a very brutal campaign against them that earns the name 'the Bloody.' He ends up king when the better-claim dies without children, and unhappily accepts. Since this is a portal fantasy exploiting concepts of the 'return to childhood' element (and this technically takes place at the end of the first novel/between first and second), what then happens is his wife - a thirty-year-old who, thanks to a botched spell and some lucky genes, ended up forced back into her ten-year-old body in another world instead of wiped out of history and her children with her - returns to that world at twenty-four - but it's been far longer, and now her eldest son is the same age as her father and she gets to witness the burden of kingship upon him. (At least that's the idea, I've stalled at the moment of her, her grandson, and all the loot she could afford and fit into nine square metres arriving in Aglan. Partially because I really ought to write the story of her and her brother growing up in Aglan returning home first.)
Of course, back in Middangard, I also have a Helen of Troy-style kidnapping and war (and it's definitely kidnapping) in order to specifically explore the impact on people after the war is won. I'm actually a big fan of 'rebuilding the world/our lives after a massive change,' as well as lords and ladies, so this might be why I quite like monarchies.
I've probably explained in far too much detail, but my simple point is that monarchies can make an extremely convenient storytelling tool. I'm also a little bit tired of people going 'but in real life!' as a complaint, because the whole point of fantasy stories is that they're that: fantasies. While I love new envisionings and concepts and examinations of the fallout of changes, I do get frustrated with people who complain that things didn't/don't work like that, because that's usually the point.
And as my last point: Rome started out as a Republic, but a Tyrant or Dictator (I can't remember which) was, if I recall correctly, someone who could be appointed in troubled times to get everything sorted and was then expected to hand back power at the end. (Tyrant might be Greek - I could be getting my concepts mixed up.) Also, one could argue that by assassinating Julius Caesar because they were worried he was going to declare himself Emperor, the conspirators may have actually paved the way for Augustus to do just that, given the chaos that followed.