You've correctly pointed out how Markov models and modern LLMs are similar in a way, and how modern LLMs don't have full agency. But this doesn't rule out the possibility that Markov models are very slightly conscious in some human-like way.
All information processing happens on matter-based substrates, brains and computers alike. I'll even provide a somewhat specific hypothesis: the pattern and temporal sequence in which particles such as electrons interact with eachother in a circuit dictated by an intelligent computer program may have physical similarities with how particles interact in a human brain, and certain patterns or temporal sequences of interaction may be responsible for consciousness. I'm not aware of any existing evidence that would rule this out. And inquiry about consciousness has always been trending downwards toward lower-level explanations over time. It might even turn out that solving the hard problem of consciousness is equivalent to finding a unified theory of physics.