Wasn't "right after"; the King is executed in 1793; Napoleon is crowned Emperor in 1804, so that's eleven years later. At the time, the life expectancy was not more than 40 years, by 1804 many with adult memories of the Revolution had already died.

More importantly for the folks still living, the Revolution had produced the brutal Terror, the corrupt Directory, foreign armies on French soil, and then finally a man with a plan, Napoleon. Long before he became Emperor, he was a heroic figure, and dictatorial figure in France-- he's the unchallenged ruler from the Coup Brumaire of 1799, and is generally popular.

It's also incorrect to call the French Revolution "supremely popular" -- it was far more accidental than the product of deeply held sentiments. The execution of the Louis XVI was a matter of deep ambivalence-- a year after the executions, there were no celebrations, and the event was subsequently the subject of a distinctly awkward public silence. There was a bloody three year revolt in the Vendée, and smaller revolts in other places, like Lyons, where the Revolutionaries, needing to execute more than could be conveniently guillotined, herded prisoners in front of cannon and blew them to pieces (this is how Fouché earns his sobriquet, le mitrailleur de Lyons = "the gunner of Lyons").

And of course the Revolution devoured many: Danton and Robespierre, the feared prosecutor Fouquier-Tinville; a butcher's bill that sapped the Revolution of both its leadership and its legitimacy. By the time of Napoleon's rise, there were still a few revolutionary spirits still in or near power -- Fouché and Lazare Carnot come to mind-- but they had long since compromised revolutionary zeal with the ambition to keep their heads attached to their necks.

There _were_ idealists, erstwhile partisans of Napoleon who were disappointed by the coronation, famously including Beethoven, you'll sometimes hear a story connecting this sentiment to the naming of the "Emperor" Concerto. That's more a story than a reality, but Beethoven's feelings were clear about Napoleon, he admired him as First Consul, didn't as Emperor.

His French Empire was exactly what the revolution had deposed: an absolute monarchy.

No, that's not how he characterized it. You may think its a distinction without a difference, but the French distinguished Napoleon from the Bourbon Kings. Napoleon was "Emperor of the French" -- eg people. The Bourbons had been "Kings of France" -- that is the State; the perhaps hair splitting point that's being made here is that he is a monarch "of the French people", a product of their will.

In a very practical sense, the Napoleonic Imperial state was not a continuation of the French Royal state. The nobility having been minimized, local political power belonged to the bureaucracy, which Napoleon reorganized according to his plan.

There's an obscure work called De L'état De La France, À La Fin De L'an VIII (On the State of France at the end of Year 8). Nominally authored by one of Talleyrand's underlings, it bears Talleyrand's hallmarks, and is ascribed to him. In it - and this is written before Napoleon is crowned-- Talleyrand makes the case that Napoleon is a special and historically unique man, embodying the nation; implicitly he earns his elevated station by service to the nation, while a King merely inherits his. So Napoleon's rule, whether as Consul for Life or later Emperor is qualitatively distinct from that of the Bourbons. Talleyrand pitches Napoleon against history's most celebrated captains, and finds him more worthy. He compares with Caesar, and is better than him:

Caesar spent twenty-five years in the fatigues of war and in accomplishing his successes before he established his reputation; the fame of Bonaparte, as rapid as it was early, was gained in less than five years.· In short, Caesar had only to contend when from Rome with generals of.no celebrity with barbarous people and nations unknown, and was indebted for his power solely to the victories he gained over his fellow citizens: on the contrary, it has been the destiny of Bonaparte to subdue nations the most warlike, to vanquish the best disciplined armies, and to overcome the first generals in Europe, and never to have fought but against the enemies of his country.

. . . point being, this is not a loafer inheriting a crown and sponging off the people, but the nation's true destined captain, this Emperor is something entirely different.

Napoleon himself made a similar distinction, words to the effect: "a man who inherits the throne is shaved by a servant, a man who makes himself Emperor shaves himself"; Napoleon was an autocrat and a monarch, but cut from a very different cloth than the kings of France, he claimed a personal and national legitimacy, not a patrimony. He also chose the honeybee as the symbol of his monarchy-- in part because "they work" (there are other reasons; a historical connection to the Merovingians, and a possibly specious observation that Napoleon's honeybee looks a bit like an inverted fleur-de-lys, the symbol of the Bourbons).

In this effort, Talleyrand was joined, with mordant historical irony, by one of the regicides, Joseph Fouché, a sometime member of the Directory who'd played a considerable part in the decision to execute the royal family. Fouché and Talleyrand were a sort of slippery odd couple, Talleyrand managing high politics, Fouché occupied with the low. They both found Napoleon's ego a convenient horse to ride, both played key roles in the Coup Brumaire and they were happy to flatter and encourage Napoleon's further ambitions. Around them were a group of what some call neo-monarchists, including Napoleon's brother Lucien, an important figure in his life and one of the plotters in the Coup Brumaire which brought him to power. Writing later, Talleyrand explained how he saw the process:

'[A] temporary sovereign had to be created, one that could become sovereign for life, and then hereditary monarch. The question was not whether Bonaparte had the most desirable qualities in a monarch ... The real question was how to make Bonaparte a temporary sovereign.

And finally, the proximate impetus for his coronation is a mirror image to the execution of the Bourbons- an attempted assassination in 1802, undertaken by Royalist agents. The bomb ("machine infernale") failed to kill Napoleon, and it produced a surge in his popularity. This was followed in 1803 by a popular war, and then a national plebiscite to make him "Consul for Life"; from which position it wasn't a long way to the imperial crown.

Sources

De l'état de la France, a la fin de l'an VIII

Napoleon and His Collaborators: The Making of a Dictatorship

NAPOLEON AND THE FOUNDATION OF THE EMPIRE

Recollections of the Private Life of Napoleon, Volume 1

LE PLÉBISCITE DE L'AN VIII ou LE COUP D'ÉTAT DU 18 PLUVIOSE AN VIII