Mr. Mic,
I think "reason" works differently - not the same as "fact" or "supposition".
"the fact that I am thirsty" is "fact = 'I am thirsty'", i.e., the 'that' clause is a noun clause in apposition to "the fact".
However, "the reason that I am thirsty" is not "reason = 'I am thirsty'", but "reason for my thirst" in the most natural reading of "reason that". Here "reason = something else not yet mentioned".
"The reason that I am thirsty is that I have not drunk anything all day."
I'm inclined to think of "that I am thirsty" as an adjective clause modifying "reason" rather than as a noun clause in apposition to "the reason". Still, there is a reading of "reason that" which is like "fact that", as follows:
"The reason that I am thirsty is not a good reason for my inattention" is "The reason - which is that I am thirsty - is not a good reason for my inattention".
In the second case I'm inclined to say the 'that' clause is in apposition. (I admit the sentence I constructed may require a somewhat forced reading, as "reason" seems nearly synonymous with 'fact' here.)
The interesting thing is that "the reason why" can substitute in the first type, but not in the second.
I wonder if this is peculiar to "reason". Is this the only noun that can be followed by a "why" clause? Well, I guess "explanation" might work, but what else?
CJ