I was under the impression that all Zod needed was
to extract the codex, and thus all of Krypton's unborn DNA sequences.
Aren't the contents of the ship superfluous?
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The codex contains the DNA of the unborn kryptonians, but he stills needs an artificial womb to grow them. The ship contains one of these devices. You can see one in Krypton, in the chamber where Jor-El steals the codex.
The ship that Superman destroyed was originally meant for colonisation purposes, as such, it contains one of these "genesis chambers" that would allow to populate a newly krypto-formed planet.
It's having all the DNA of your people in a set of floppy disks and not having access to a reader.
Yahoo Movies did a Q&A hosted by Kevin Smith with Zack Snyder and Amy Adams in the studio and Henry Cavill joining via satellite for the Blu-Ray release of the film. In that video, which is still available online, Snyder was asked a question about the Codex by Steve Younis (of SupermanHomepage.com) who indicated that while the base genetic information comes from the Codex, the variation and programming arise from the Genesis Chambers.
Zod is preeminently preoccupied with eugenics. He genuinely believes that Krypton needs the "right" bloodlines (non-degenerate) to go forwards... a belief bred into his blood, potentially somatically programmed into him as a trainee, and reinforced by his own will. He believes it so strongly he literally calls natural birth heresy. He believes in it so consistently that despite 33 years passing at no point did he mandate natural reproduction on the Black Zero OR even the beginnings of cloning by Jax-ur.
Zod's vision of Krypton is narrow and orderly and specific. He would rather die that see a Krypton filled with super-powered citizenship. He risks his liberty and overthrowing his government to ensure eugenic purity. So although the Codex would have provided enough information for reproduction... he already had that in the form of a mixed-sex crew and a brilliant scientist / geneticist aboard his ship... rather, Zod required that Krypton conform to that vision that he had in mind when he risked upsetting the Status Quo with his coup. One where Krypton's bloodlines were selected and dictated by himself... not random chance, not a limited gene pool of clones, and not without the guidance and control of a Genesis Chamber.
It may seem a narrow and inflexible view... but that's how radical fanatical individuals can be!
The corollary: "Why then isn't the Scout Ship enough?" From the Prequel comic we know that each Scout Ship was equipped with a Growth Codex (but not, necessarily paired with a World Engine, again suggesting that only planets that were within the band of habitability without a World Engine could even be terraformed to begin with). However, as Zod's vision of Krypton is driven by eugenics, the Codex within the Scout Ship would not have the benefit of the past 18,000 years of refinement and would be as much heresy for Zod to adopt as any other reproductive measure.