The Republic is structured in a way that supports and strengthens the need for philosopher-kings, because through them the Kallipolis is brought to life and genuine happiness of society is achieved. It is important to recognize that Plato understands his utopia only as a concept but not as a political reality. Therefore justifications of Plato's thought do not have to be adapted to the reality of Greece but to the reality of the "retained in heaven" (592b) Republic. Hence, it would be false to say that the arguments put forward by Plato are not persuasive. I will expand on these thoughts based on the definition of justice, and the formation of philosophers to create and rule the Kallipolis.
Plato's main reason for righting the Republic was to counter or improve the Athenian democracy which at that time was unstable, leading form anarchy to tyranny (324b-325c). However fundamentally Plato believed that the cause for this instability was an outcome of no "competent leadership" (501b) with "proper skills and morals" (karpowicz1). The Republic therefore in the first couple books looks at the definition of justice and what should a just state look like to prevent tyrannies of Athenian democracy. It is this question which makes Socrates wander form person to person asking "what is justice" (books I-IV)? Unfortunately the definitions given by Cephalus, Thrasymachus, and Polemarchus are insufficient because each one of them is only an opinion. Therefore Socrates concludes that justice "concerns no ordinary topic but the way we ought to live" (352d). And if we ought to live by justice which is virtue and goodness in an organised politically state concerning everyone's interest (however not equally as stated by famous critique Karl Popper), we need philosophers who understand this concept knowledgeably.
By saying: "the way we ought to live" Socrates is directly pointing to the world of forms (472-483) and the theory of the good (508c-509a). Plato's intention in the Republic is to show that justice does not have to be "what is advantageous for the stronger" (338c) or to "to treat friends well and enemies badly"(332d), but a achievable status of a city given that philosophers rule. It should be understood that philosophers because of their nature as wisdom lovers (475b8-9) are capable of identifying the form of the good, which is true knowledge not based on opinion (480a11-12) but on concepts that are "free from the distorting influence of power or ideology (p64 big book). Because philosophers can distinguish between true and false belief, they are master theoreticians/practitioners who can heal the society of its illness best described by the pessimistic and corrupt definitions of what justice is by Socrates' interlocutors. Plato believes that philosophers knowledgeable rule in the name of justice and virtue, will guide its society "to proper ends" (karpowicz2).
The other part to the question "why should philosophers rule" lies strictly in their education and privileges, because how can it be possible to create such human beings with a inhuman grasp of knowledge far superior than anyone else (484a-487a). However it is necessary to note that it is not the education itself which makes the philosophers good candidates for kings but rather the aim of education which is to create a just and truth seeking person who understands and sees the good (501b, 517b). Therefore the philosopher must not love one part of knowledge but all of it (474c8-11, 474c6-6), in order become virtuous. Nevertheless the main question arises when we try to think logically about all of this. Why is it that someone would spend so much time educating themselves, almost 50 years (537b2-6, 537b8-c3, 537d3-539e2) in order to access the forms and the good, given that he also would lose his privileges to private property and be forced to eat in common dining halls and share wives and children in common (C. D. C. Reeve). The answer that Plato gives to back up his vision is that after 50 years of education a philosopher would no longer rely on "honour seeking" but would take pleasure from wisdom and truth (580a-592b). That way he would not become the master of his citizens but a leader/ally (417a-b). As said by Professor W.J. Korab-Karpowicz, "Plato assumes that in a city where rulers do not govern out of desire for private gain, but are least motivated by personal ambition, is governed in the way which is the finest and freest from civil strife" (520d). Otherwise ruling for the philosophers would not be enjoying but rather seen as something necessary. Similarly can be said about the banning of private property (485d10-e5) since according to Plato without this ban it would be like taking a river and dividing its main stream into many smaller streams (485d6-8). The metaphor means that by banning economic property philosophers are not distracted from their main focus, ruling, but rather forced to concentrate on what they are the best at.
All of this brings us to the wider spectrum of the Republic, meaning usage of philosophers to rule justly in what's known as the ideal city or the Kallipolis. The question that Plato encounters relates specifically to the "allegory of the cave" (514a). How do you show the people how to live a virtuous life in a community and practice the good? More specifically how do the money lovers and honour lovers experience the good given that the philosophers are out of the world of shades? The answer lies in the formation of the kallipolis which as Plato proves is in fact a real possibility (68 big book). By forming the first polis in which the money lovers enjoy their life maximally, the second polis is created to manage the appetitive desires of the money lovers. The second polis if best fitted for honour lovers who get to practice their spiritual desires, by rule the polis through true belief. However both in the first polis which then converts to the second polis consisting both of money lovers and honour lovers, still does not allow its citizens to experience the good. Justice is not practiced to its fullest extent because there is nothing to counter the honour lovers (C. D. C. Reeve). Due to this the third polis is introduced with the philosophers as the fully virtuous individuals who allow the polis to become fully just and realise its good. That way the kallipolis is formed allowing money lovers, honour lover and wisdom lovers to practice the good and live a virtuous life which in the long term equals genuine happiness as suggested by Christopher Bobonich (Platos Utopia Recast p216). Philosophers through their access to the good and access to true knowledge of the forms best explained by the "metaphor of the sun" (509b), form a just city, specifying what's just and what's unjust for the producers and guardians. As stated by C.D.C. Reeve by appropriately modifying the psyches of the members of polis to form a properly ruled kallipolis by the philosophers, allows the people to become complete, bringing the polis into existence.
With the creation of the Kallipolis, people inside it keep their psyche ultimately happy. However there is a question to be asked. The metaphor of the ship states that a true pilot out of necessity must constantly look at the stars to know the ships direction and guide it properly, as a part of his craft. However the people look at the pilots as "stargazers" who do not satisfy their desires and do nothing on the ship (CDC Reeve). According to Plato people do not understand what ruling is about posing a threat to philosophers, thus proving that the system of the kallipolis in truth might be a complete failure. If people discard philosophers, the polis triplet breaks down, causing the public to elect leaders who promise the most in the short time (487c7-489c10) leading to the re-rise of tyranny and the loss of the public good. So how do we convince the people to trust the philosophers in their rule? According to George Klosko, Plato does not have a precise answer to this question other than influencing the ones in power. Aristotle critiques the philosopher kings stating that their rule is based upon achieving a complete utopia in a society (Anton-Herman Chroust: Aristotles Criticism of Plato's "Philosopher-King"). However, Aristotle does not understand that a utopia in fact can be achieved. Plato suggests that rulers might find it useful sometimes to lie, the best example being the noble lie of the "Myth of the Metals" in order to form love between the citizens and maintain Platonic triplet state structure (415d3-5). By doing this the philosophers justify that the society must be divided into different groups, because each citizen is a representation of a specific metal. Therefore if the citizen understands his role in the society, he should understand the role of philosophers in the society as well. With this the citizens realize that philosopher kings should rule. If the noble lie works, then the Kallipolis based on the convincing description put forward by Plato, is a real possibility. Therefore not implementing philosophers would result in a loss of good to the society, given that nothing else is in their way (502b).
My point through the essay was to indicate that Plato's conclusion that philosophers should rule is in fact persuasive and realistic. For every question to counter philosophers rule, we find a rational defence to prove the correctness of Plato's theory which "perceives the means necessary for the desired end" (George Klosko : The developments of platos political theory, page 191). However, Plato makes it clear that it is just an theory but not a easily achievable political reality (592b) and in truth there is now way how it can be brought to life (592a-b) other then with the use of power/force best put by Machiavelli: "Thus it comes about that all armed prophets have conquered and unarmed ones failed" (The Prince, ch.6: trans. L. Ricci and E. R. P. Vincent, New York, 1950).