The incompetence of the Republic for governing its Empire can very broadly and roughly be divided into two aspects:
Firstly; the army. Viewing the events of the last century of the Republic’s existence, it is pretty hard to argue with the conclusion that the military’s habit of following influential generals into battle against their fellow Romans was a problem that had to be addressed. A military autocracy of one kind or another was inevitable – Caesar recognized this.
Secondly, and even more importantly, IMO, than the military issue, to say that the Republic was doing a fine job of ruling itself is a blatent contradiction to the facts. Plundered and Oppressed Provinces, young noblemen with bad habits of running into debt, massive unemployment and the general destabilization of the economy related to the Latifundia and the massive use of slaves, gangs ruling the streets, an oppressive oligarchy that was prepared to fight to the death (literally) against any attempt at what the system needed more than anything else: reform. Gelzer rightly points out that the problems that Caesar faced upon the conclusion of the Civil War (the economy in shambles, widespread devastation and terror, etc) had by no means been caused by the Civil War that had just been fought: they had been brewing for a long time, and all the Civil War had done was bring them over to boil. There is no greater eveidence of Caesar’s sheer sanity above all else that having gained power he did his utmost best to use it intelligently to try and improve things, rejecting both populist and optimate interests alike in favor of what was the greatest benefit to everyone in general.
That is not to say that the Republic could not have continued to rule its empire feasibly for a few more years, without change. Of course, the problems mentioned above would inevitably have brought the whole system crashing down, and we today would certainly not remember Rome for being the great civilization that it ultimately became. But it was the high mark of Caesar’s statesmanship that he recognized that it did not have to be like this; indeed it should not be like this. Why should the empire remain a series of provinces at the mercy of a city-state steadily sucking them dry (this is not an exageration: the marked economic decline of the areas under the rule of the Roman Republic over the years is a fact – it was not until the Empire that they recovered) when instead it could become a universal Romanized Empire with wide-spread benefits for many of its 60 million odd inhabitents rather than just the ones who lived in Italy? There can be no greater proof of the soundness of Caesar’s vision than the results of its being implemented. It was the transformation of Rome from a city-state system into a wide Imperial Nation-State that lead to the prosperity of the Pax Romana and the real flowering of Roman Civilization.
Of course, to achieve this, the Oligarchic Republic had to go. And in many ways, good riddance. The rule of the Emperors was, many argue, not much better, and this is one of the reasons why I see Caesar’s demise as a tragedy. He had succeeded in creating the Empire, but it was left to a still brilliant, but nevertheless lesser man – Augustus – to devise the way it would be ruled.
Unlike Augustus, who purged the old nobility and broke it forever, Caesar recognized that while the old nobility might have been corrupt, they nevertheless carried with them the traits that had made Rome great in the first place. This was one of the key stones in his policy of clemeny and reconciliation: he recognized that the new Nation-State coupled with the old strengths that had lead to Rome’s rise, freed from their antiquated principles, would be a potent force indeed. And while at it he also intended to try and improve life for the lower classes. As was, Rome had to make do with the Nation State, and the lives of the lower classes were improved.
An autocracy was required, and now for your question of: Why Caesar? Pure and simple because there was nobody better suited to the job, and he knew it. His career firmly established him as a believer in meritocracy, and meritocracy in a nutshell is that the best man for the job gets the job. Caesar had many admirable traits, but modesty was not one of them. He knew he was the best, and that was why he won out in the end. Gelzer sums it up nicely: Caesar was conscious of his power to rise to become master of an Empire that he could reshape in his own image. His efforts in Gaul confirmed his rank as the First Man in Rome, the Civil War that followed represents the defence of his Dignitas.