They learn during training, which is what we're talking about.

They already learned. A lot or basically everything evern written and available digital.

And context window work very well. You can 'teach' an llm a new programming lanuage and other things through it.

They do learn in context, and very sample efficiently. Continual learning is active area of research and we sort of already have something resembling it with persistent context. So yes they do learn.

The LLM itself doesn't, but agents can research, compare, add to their memory, and use that to narrow the results down to a probabilistically higher set of outputs; I have used an LLM for my own MRI results and it was nearly spot-on, verified by a subsequent visit to a specialist. YMMV as they say. But I do believe we are entering the era where LLMs are considering past interactions and long context windows to inform it of personal preferences and history in order to output more accurate results.