It's easy to say that, but "surely it must be possible to connect an llm in such a way that it becomes intelligent" (tell me if I'm misinterpreting) is not a demonstration of anything. It's basically restating the view from the 50s that with computers having been invented, an intelligent computer is a short way off.
The network has learned human patterns of language, knowledge and information processing. If you want to update that, you can re-train it on a regular basis, and re-play its sensory/action history to "restore" its state.
If you mean "learn from experience", (1) a lot of that is pointless because it's already learned from the experiences of millions of humans through their writing and (2) LLMs can "learn" when you explain consequences.
In theory they could learn by having their discussions fed back to them in the future, and it does seem that this occurs.
Now, there is no continuous learning in the human/animal sense. Of course it is thought that even humans have to sleep and re-weight their networks so short term knowledge is converted to long term knowledge.