Why is Sir Arthur Conan Doyle used as a character to explain the expertise of Sherlock Holmes in "The Red-Headed League"?
I think the author, in the persona of Dr. Watson, acts as a "foil" to Sherlock Holmes in all the stories. This is a literary term that has two meaning. First, a foil may be present to "offset" the character of the protagonist, in this case, Holmes. Second, the foil might act as device to somehow "spoil" the actions of the protagonist.
In this case, Watson is a foil in the first sense of the term. Without Watson's inability to think as Holmes thinks, would we be able to appreciate Holmes' genius? Probably not! It is Watson's lack of imagination and the inability to reason deductively that allow Holmes to shine.
The other way to view the character of Dr. Watson is as the comedian's "straight man." In a comedy routine, sometimes there must be someone to ask the questions that will elicit the comedian's punchline. In the Sherlock Holmes stories, much of what we learn is through Watson's questions to Holmes. This allows Holmes to deliver the brilliant answers to someone. Otherwise, we would simply see Holmes talking to himself!