The logic is simple.
No, your logic is flawed, you are using the presumption of conscious control to try to demonstrate conscious control, it's a circular argument.
Time related cause and effect reactions are what they are - uncontrollable reactions cannot provide the control needed to drive rational thought processes to reach consciously verified conclusions
We don't 'consciously verify' anything. We become consciously aware of that we have subconsciously concluded that a given argument is valid - we could be wrong, our logic could be flawed, we might have erroneous precepts, whatever. The conscious awareness, though, happens after the thoughts have already been thunk.
There is no feasible explanation for how rational thought processes can emerge from uncontrolled material reactions.
On the contrary, if they are rational they flow from the evidence and the precepts, that's entirely without a need for a conscious overseer. What is the case is that there's no feasible explanation for how something can be both 'will' (i.e. a decision made upon consideration) and 'free' (i.e. not controlled by outside events) given that we are the product of outside events.
In order to try to rationalise this you need to invent things like non-material, undetectable, elements of external consciousness which unobtrusively interact with the brain because otherwise you have a conclusion you don't like - it's an argument from incredulity. It's not that the model depicted is not feasible, it's that you can't accept the conclusion and its implications.
O.