Much earlier in this discussion - and apparently forgotten - are natural codes such are tree rings, varves, and Fraunhofer lines. Codes can be natural, and if these things are not deserving of the label “code” then neither is the genetic code.
The problem, as with most arguments for ID, is the lack of specificity. It’s a hasty generalization from the specific example of designed code to the universal claim that all codes are designed. As we see in every discussion of Information and Coding (like this one), the argument for Design evaporates when careful definitions are applied.
More generally, making material arguments for the supernatural is just begging for someone to come along and poke holes in that argument. It happens every time.