Modified from an earlier answer

There is an important step missing in this formulation: the Consulate. Far from a break with prior Revolutionary politics, this limited dictatorship was an evolved out the political culture of the post-Terror France. As the extreme period of the Revolution passed, many of the surviving politicians began to speak ill of too much democratic rule and freedoms. The attitude of various Revolutionaries towards executive power was always very unclear, but there was throughout the Revolution voices that called for strong central order. The experience of the Terror made many within the Directory leery of pure democracy (Cabanis said democracy was "odious to all sensible and right-thinking men" and "kept all citizens in a constant state of agitation and anxiety") and to seek an ordered solution to political power. Although subsequent Napoleonic propaganda would emphasize him as the natural and inevitable choice for the Brumaire coup, there were several other French generals that various factions courted for a coup. Emmanel Sieyès, one of the key plotters in Brumaire, would famously quip, "I need a sword," in other words, a man who could command the loyalty of one of the few powerful institutions that possessed a great deal of popular legitimacy within the Republic, the army. Bernadotte, a general and Minister of War, was the favorite of the Neo-Jacobin factions that wanted to reestablish the Revolution's leveling tendencies. Jean-Baptiste Jourdan toyed around with many different factions, but never made a firm commitment. Lazare Hoche, who enjoyed almost as much fame as Napoleon had unfortunately died of tuberculosis before Sieyès's initial soundings towards him could bear fruit. Jean Moreau, the youngest general in the army, refused to compromise himself with politics despite entreaties to do so.

The path to the empire from the Consulate was a very gradual process between 1799-1804 which meant that Napoleon possessed a degree of flexibility when it came to his self-coronation. There were two trends at work in France at this time which help explain why this transition was relatively smooth. Firstly, Napoleon greatly benefited from and actively encouraged a depoliticization of the French government that had begun under the Directory. The Consulate's emphasis upon the power of executive authority greatly helped in this manner as it became much more difficult for the Tribunate or the Senate to form an alternate power bloc against the First Consul. Napoleonic censorship also denuded the republicans remaining within the French government of any organs which to voice opposition. The Consulate did witness some friction between Napoleon and the other branches of government. For example, Napoleon's partisans within the government had to work quite hard to get the various organs of state to pass much-vaunted Légion d'honneur, which the remaining republicans in government saw as an affront to the history of the Revolution.

Napoleon responded to opposition with a measure of tact that mixed with an iron determination to get his way. Napoleon rewarded his loyal servitors with new titles and estates. He also though expected these servitors to exhibit a degree of competence and was a very stern taskmaster for the running of the French state. Napoleon also acted quite differently than other French executives towards opposition. Lazare Carnot, one of the unimpeachable heroes of the Revolution, exemplifies how Napoleon defanged his opponents. Carnot was one of the stalwart foes of Napoleon's accumulation of power. When he voted against the making Napoleon consul for life in the Tribunate, he had it put into the official record that by voting no, he had written his own condemnation. In prior regimes, this would have been a death sentence or exile, but Napoleon allowed him to retire and function as a private individual in the empire. LeBrun, the third consul and the most republican of the trio, similarly was also allowed to play a less active role in French politics, while still receiving titles and rewards. The other consul, Cambacérès, became Napoleon's right-hand man within the imperial state and also was a beneficiary of Napoleon's largess. In short, the French government became both pliant and reasonably competent by the time of Napoleon's coronation.

The second trend at work in France at this time was the image of Napoleon himself. Napoleon enjoyed an unprecedented degree of popularity during the Consulate. The period of 1800-04 was a time when Napoleon scored a number of diplomatic coups that greatly added to his prestige. Even though it was short-lived and fundamentally unworkable, the peace with Britain was an accomplishment that no prior Revolutionary government had accomplished. The Concordat with the papacy also seemed to be ushering in a period of domestic peace within France. This is why the 1802 plebiscite for a life-consulship received a broad base of support, even if the Napoleonic state had its thumb on the scales. The public image of Napoleon also underwent a profound transformation at this time. Napoleonic artists tended to less depict him as a republican general and much more as a regal figure who employed the gestures and symbolism of monarchy. As with the government, Napoleon rewarded loyalty to the increasingly personalist regime amongst the public while side-lining any opposition. Fetes, public festivals and new forms of imperial nobility had a strong element of a cult of personality about them, but they also were enjoyable distractions.

The question as to how deep this support for Napoleon ran both in the government and the public though is an open one. While Napoleon was a popular figure, the expressions of popular support such as petitions to name him a hereditary emperor were definitely manufactured. For instance, there was a clear connection between the army's leadership drafting these petitions for the ranks to sign. The empire's fetes surrounding the coronation were large, but a number of observers noted they were not particularly boisterous. There was also a great deal of apprehension about the hereditary nature of the empire within governmental debates. The issue of succession was far from clear in case Napoleon died or was otherwise incapacitated. Although Napoleon's nominal heir was the son of his brother Louis and step-daughter Hortense, the succession issue was something quite vexing to the French government. Underlying this anxiety is a fear that the entire imperial edifice rested upon the shoulders of one man, Napoleon.

Herein lies the double-edged nature of Napoleon's path to power. He constantly portrayed himself as an extraordinary individual who operated outside the limitations of normal human beings. During his imperial coronation, Napoleon did not have the pope crown him, but instead he crowned himself in the presence of the Pontiff, a move that signified he was the active party in the coronation. The justifications used for the transition from a consulship to a hereditary emperor explicitly stressed this. But this type of rationalization ensured that it was much harder for anyone to follow him. So in the end Napoleon, much more than Louis XIV, encapsulated the dictum "I am the state," which ensured that Napoleonic empire was hostage to its emperor's personal fate.

Sources

Bergeron, Louis. France under Napoleon. Princeton, N.J.: Princeton University Press, 1981.

Dwyer, Philip G. Citizen Emperor Napoleon in Power. New Haven, Conn: Yale University Press, 2013.

Woloch, Isser. Napoleon and His Collaborators: The Making of a Dictatorship. New York: W.W. Norton, 2001.