Humans do not have a continuous stream of thought when they are asleep, even if their brain is still doing things. Your original example (the LLM can't take actions between problems) is literally the same as the fact that the human can't take actions while asleep.
Of course, nobody has a clear enough definition of "sentience" or "consciousness" to allow the sentence "The LLM is sentient" to be meaningful at all. So it is kind of a waste of time to think about hypothetical obstacles to it.
I'm not sure we always have a sense of time passing when we're awake either.
We do when we are focusing on being 'present', but I suspect that when my mind wanders, or I'm thinking deeply about a problem, I have no idea how much time has passed moment to moment. It's just not something I'm spending any cycles on. I have to figure that out by referring to internal and external clues when I come out of that contemplative state.
> It's just not something I'm spending any cycles on
It's not something you are consciously spending cycles on. Our brains are doing many things we're not aware of. I would posit that timekeeping is one of those. How accurate it is could be debated.
Of course, nobody has a clear enough definition of "sentience" or "consciousness" to allow the sentence "The LLM is sentient" to be meaningful at all. So it is kind of a waste of time to think about hypothetical obstacles to it.