Obviously if it can't improve its code then it can't improve its code. But you seem to think there's some barrier that means that even if it was a better-than-human programmer (implicitly for general-intelligence reasons), it wouldn't be able to geometrically self-improve. And it's not at all obvious (at least to me) what that barrier is. We can already throw ChatGPT or equivalent at a codebase, ask it to suggest improvements to the code, and it does; the hit rate isn't great, but it's already good enough that people are incorporating it into their workflows.
As far as I can see, that's enough. Yes, people who are worried about this don't spend a lot of time going through the details, just as people who are worried about the military staging a coup don't spend a lot of time going into how they will get weapons or coordinate their movements or what have you. Those things are pretty complex, but they're ultimately routine; we are right to gloss over them, and if someone thinks that those things make a coup impossible, the burden would be on them to explain why (maybe there's some specific protocol that means they can't get access to weapons without authorisation from the legitimate authorities - but you'd need to explain what that protocol actually is and justify why you expect it to be wartertight).
Your argument seemed to be that babysitting the kind of complex code that we use for AI is not merely complex, but so complex that it requires some unique human ability that we would not expect a ChatGPT-like AI to possess (implicitly, that it requires something a lot more difficult than making improvements to code). And I just don't see that. Put it this way: we would never expect a human worker at these companies who was smart enough to make improvements to the core AI code to get stuck when it comes to actually running it. So why would it be an issue for the AI? I do think the burden is on you here given that the whole premise is that the AI is smart enough to improve its own code (you're welcome to argue that it's impossible for an AI to ever get smart enough to improve its own code - but in that case all this talk about operational complexity is beside the point).