Bonaparte's promulgated the Civil Code because he happened to be Prime Consul when the Civil Code was achieved, but his influence on it was somewhat minor (albeit real: few conservative considerations here and here). The Civil Code redaction was, mainly, Cambacérès direction's work. Cambacérès worked on it since the 25th of June 1793, in the early days of the French Republic, under the National Convention. When Bonaparte made his coup to become Consul, then Prime Consul, the French Code Civil was quasi achieved.
The personal link between Bonaparte and the French Civil Code was mostly propaganda, albeit it can't be denied that the authoritarian regime he established contributed to make its adoption easier: it was done, without much opposition, thanks to the constitutional "bug" used by Bonaparte to fire tribunes of the republican and liberal opposition from the Parliament.
You're downplaying the role Napoleon had in getting the code implemented.
It's true that the National Assembly unanimously recognized the need for new a new code of laws in 1791 and the National Convention organized a commission in 1793 to draft said code under the leadership of Cambacérès, before Napoleon's accession. However, the first draft was rejected on the grounds of it not being accessible to the public. So Cambacérès prepared a second draft and it was introduced in 1794. But once again, it as rejected. So Cambacérès once again drafted a new code. But even after three series of revisions, the code still failed in 1796. And when a fourth code was introduced by a reorganized commission in 1799, it was once again rejected. Despite four separate attempts at passing the code, the ruling body couldn't agree do it. This shows how fractured and ineffective the ruling body of France as in a span of eight years, they had accomplished nothing of note in this regard.
This is why Napoleon was vital. Upon his Consulship in 1799, he established a new commission of jurists to write the codes in 1800, with Cambacérès included. Given that most of the work was completed at that point, the code was completed in 1801, and by 1804, the code was passed with Napoleon actively supporting it.
This shows that without Napoleon's involvement, the code would have taken much longer to pass (if not at all), and might not have passed in full. While it's true that Napoleon's conservativism had an influence on the final draft, the code being implemented alone made progress by cementing ideals of the revolution into common law. Its also still important to note that because of France's expansion under Napoleon it was spread beyond France and into Europe.
These are the series of events as I understand them. I'm fairly certain that I did the right research into this response, but if in wrong in some aspect, let me know.
