Why has no country ever tried to emulate the succes of the roman republic, and why did it take european countries so long to get rid of their kings when all it took the romans was one bad king?

Though not an exact emulation, both the French and American revolutions were directly influenced by the Roman Republic and the ideals of Libertas achieved by the early Romans, as Sellers writes (reference at the bottom):

“A new "Senate" would meet on the "Capitol" hill, overlooking the "Tiber" river (formerly "Goose Creek"), as in Rome, to restore "the sacred fire of liberty" to the Western world. The vocabulary of eighteenth century revolution reverberated with purposeful echoes of republican Rome as political activists self-consciously assumed the Roman mantle. James Madison and Alexander Hamilton, the primary authors and advocates of the United States Constitution, wrote together pseudonymously as "Publius" to defend their creation, associating themselves with Publius Valerius Poplicola, founder and first consul of the Roman Republic. Camille Desmoulins attributed the French Revolution to Cicero's ideal of Roman politics, imbibed by children in the schools. At every opportunity, American and French revolutionaries proclaimed their desire to re-establish the "stupendous fabrics" of republican government that had fostered liberty at Rome. The Roman name of "republic" evoked first and above all the memory of government without kings. Roman authors dated their republic from the expulsion of Rome s last king, Tarquinius Superbus, and mourned its fall in the principate of Augustus.

As French and American politicians came increasingly into conflict with their own monarchs, they found a valuable ideology of opposition already fully formed in the Roman senatorial attitude towards Caesar and his successors. The guiding principle of this republican tradition, as remembered (for example) by Thomas Paine, was government for the "res-publica, the public affairs, or the public good," perceived as naturally antithetical to monarchy and to any other form of arbitrary rule. Paine and other eighteenth century republicans viewed the individual and collective well-being of citizens as the only legitimate purpose of government. Their rallying cry of "liberty" signified subjection to laws made for the common good, and to nothing and to no one else. Statesmen traced this principle to the frequently cited passage in Livy that attributes the liberty of Rome to Lucius Junius Brutus and to his introduction of elected magistrates into Roman politics, constrained by the rule of law”

As Sellers suggests, it is not that they directly tried to “copy” exactly the Romans (though in many ways they did - hence for example the movement itself being called Republicanism!) but that the foundation of the Republic served as an ideological benchmark through which they could express their desires for liberty and self governance. This is why for example during the time of the French Revolution there was an increase in paintings and art that recalled famous events from Rome’s early history (Jacques-Louis David is perhaps the most famous, such as the Oath of the Horatii). Both in France and America, intellectuals were consciously linking themselves to the Romans to both help justify and inspire their contemporaries into moving away from monarchy and towards Republicanism and they saw the Roman Republic as both their symbolic and literal predecessors.

Though not as overt, this can be seen in the artistic and intellectual culture of the English Civil War. For example John Milton’s “Paradise Lost” draws mainly from the inspiration of Virgil, who’s often-conflicted portrayal of Aeneas as an effigy of Rome’s first emperor Augustus has led many scholars to believe that Virgil’s work is more subversive of Augustus due to Virgil’s own conflicted opinion of one-man rule under the newly established Principate - rule by the “First Citizen”. It is perhaps unsurprising then that Milton’s Paradise Lost begins with Satan, cast out from heaven for his rebellion against God, the king of heaven. Some of Satan’s sentiments (“better to reign in hell than serve in heaven”) reflect a rejection of absolute monarchy that was beginning to take hold within western intellectualism.

As to why it took so long for more modern countries to reject monarchy I am no expert, but the enlightenment was built off of the back of the Renaissance - which was a “rediscovery” of Greek and Roman ideals - so you could argue that it is no accident that rejections of monarchy began to take root directly after European intellectuals began to study Rome’s republic (as well as Athenian Democracy and Sparta’s mixed constitution) in a more widespread way.

As to why Rome rejected their kings this is still a source of debate, as Livy (the main source for Rome’s early history) acknowledges himself that the stories passed down are more “poetic tales than history” and so most of the stories are more than likely legendary fables. My own guess based on studying/teaching the Roman King’s and Rome’s early republic is that there was a cultural shift within Rome that had begun before the rejection of Tarquinius Superbus. For example Rome allegedly always had a Senate, and under the Etruscan Kings there was an increasing importance placed on the increased political participation of Rome’s citizens. For example the 6th King Servius Tullius reorganised Rome’s citizens so they voted according to their property value - landowning citizens had more of a say in the political life of Rome but the burden for military service was placed on them. When the King’s were thrown out we are told by Livy that Brutus and the first republicans decided not to change too much of Rome’s constitution, but instead to dissolve the power of the Kings and divide their roles among the pre existing political bodies, such as the Senate and Popular Assemblies. Livy also tells us that the earliest Kings used to be elected, rather than it being a hereditary monarchy in the strict sense of the world. Therefore it was not an absolute monarchy like many medieval kingdoms.