Unfortunately, that argument applies to you, yourself. I mean, presumably you know that you yourself are intelligent, but you must take it on faith that everyone else is. We all could just be a kind of Chinese Room, as far as you know. Communicating with us is not a sure way to know whether we are "really" sentient because we could just be automatons, insensate but sophisticated processes, claiming falsely to be just like you.
> the LLM will assign some probability to every plausible completion -- so if you sample enough times, it will eventually say, e.g., "Well, I am not sentient."
Perhaps so. I think the mistake is trying to split that hair at all. According to BF Skinner we are all automatons, and any sense of self-awareness is an illusion. Some psychologists and animal trainers have found find that model to be quite well explanatory for predicting observed behavior. Is it correct? We will never really know for sure.
So, if a skeptical, knowledgeable user guardibg carefully against pareidolia encountered a chatbot that is sufficiently sophisticated to seem sentient to that user, it's tantamount to being sentient. For all practical purposes given our existential solitude, an entity that convinces us of its sentience is sentient, irrespective of any other consideration.
Your example implicitly acknowledges that. If LaMDA would make such an elementary error, it must not be sentient. Conversely, if it did not make such errors, it may be sentient.