Nationalised LLM? As long as the state doesn't decide what information the LLM shares (from an output and privacy perspective).

Agreed, a LOT of UBI advocates gloss over the "B" in UBI. If AI increases human productivity overall, the only morally acceptable outcomes (imo) are that everyone's standard of living increases (or at least is the same without having to work) and wealth inequality decreases (if AI is doing ~all the work, there really isn't any sensible justification for some people having significantly more wealth than others). Frankly anything else seems like a recipe for massive social instability.