It's probably helpful in this discussion to make a difference between two definitions of reasoning:
1. phenomenal reasoning, requiring consciousness and subjective experience
2. functional reasoning, transforming premises into conclusions using logic
I think you are attacking this using definition 1, whereas the article is obviously aiming at a different type of reasoning, and trying to formalize what is actually going on. It seems to be a genuine effort.
>1. phenomenal reasoning, requiring consciousness and subjective experience
I think it is incumbent upon anyone arguing that something does not posses any given property to provide a non-circular definition of what it is that they are declaring an absence of.
All of the descriptions of experiential reasoning are usually defined in terms of rephrasing of the claim "true understanding", "conscious", "aware", "knowing" all hinge on a synonymous aspect of the words that try and shift the responsibly of explanation to the next term used in a cyclic manner.
For the weaker sense of reasoning, there simply isn't any argument that it is not happening. A calculator can perform the weaker sense. The analysis of this aspect of LLMs is purely a question of how, not what.