The false conclusion that's being drawn is "therefore LLMs could not be good models of consciousness" (consciousness being a cognitive capacity). Plus, I suppose a subtle implication that a good model of consciousness is not actually conscious. To which I would invoke the spirit of the Turing test: if you can't tell the difference, is it not more sensible to say that it is.
> (consciousness being a cognitive capacity)
I don't think it makes any sense to say that consciousness is a cognitive capacity. Cognition is one of many qualia that compose the experience of consciousness from the inside, but it's not the only one, and I can easily imagine consciousness without cognition at all.
So I don't think it's weird at all to say that LLMs can be good models of some cognitive capacities (particularly the ones embodied in language) but lacks others, and overall lacks consciousness.