An LLM cannot "become" an OS. It can have an OS added to it, for sure, but that's a different thing. LLMs run on top of a software stack that runs on top of an OS. Incorporating that whole stack into a single binary does not mean it "becomes" an OS.
And the point stands: you would not write a new OS, even to incorporate it into your LLM. You'd clone a BSD (or similar) and start there.
I don't think you're getting the main point. The only application that this physical device would run is ChatGPT (or some successor). You won't be able to install other apps on it like a normal OS. Everything you do is inside this LLM.
Underneath, it can be Linux, BSD, Unix, or nothing at all, whatever. It doesn't matter. That's not important.
OS was just a convenient phrase to describe this idea.