Suspension of disbelief doesn't have to extend to the in story logic itself. They told me their setup then failed to utilize it, and in fact went with actions that directly ignore it.
It's bad writing, plain and simple.
I wish for spoiler tags cuz reasons.
The codex was "stored in his cells". That's good because to use his DNA to store other DNA would mean his DNA was possibly millions of times larger than needed, and mostly turned off, if there was a possibility of data compression. Cuz to store IN DNA you would write with DNA. So I assumed some nanocomputer or data crystal or whatever they used, copied into all cells.
I admit I am assuming that it's a whole copy every time, as I think there is way too much chance of loss due to injury and biological turn over. I can't think of any way that it is in his cells, unless it was meant that the codex was hidden in another dimension, and the key was made to be his bioprint or soul or whatever. They didn't say that.
I don't care that he is Superman otherwise. The same story logic could be used in any story featuring an alien from a culture capable of storing a civilizations worth of unique DNA inside a persons cells in any way.
It's lazy. Its a way to say Zod is a dick, but he didn't choose to have to go after Kal-el to save his species from extinction. Jor-el made sure that would be necessary.