Watson wrote:
He was a man of about fifty, tall, portly, and imposing, with a massive, strongly marked face and a commanding figure. He was dressed in a sombre yet rich style, in black frock-coat, shining hat, neat brown gaiters, and well cut pearl grey trousers. Yet his actions were in absurd contrast to the dignity of his dress and features, for he was running hard, with occasional little springs, such as a weary man gives who is little accustomed to set any tax upon his legs. As he ran he jerked his hands up and down, waggled his head, and writhed his face into the most extraordinary contortions.
"What on earth can be the matter with him?" I asked. "He is looking up at the numbers of the houses."
"I believe that he is coming here," said Holmes, rubbing his hands.
I always find it funny when Watson gives such a detailed description of a person and their behavior and then he somehow can't figure out the man is looking for their address.
I think this kind of stuff is the main source of the biggest characterization of people when it comes to Watson: the idea that he isn't observant or not very smart.
I think a consequence of having Watson as the narrator is that we have him as the
observer as well, which has interesting effects on how facts are given to us in the story.
It's generally agreed that in a good mystery the writer gives the reader all the clues without
cheating and pulling the solution out of his hat with facts we never got. Usually this is done through a 3rd person narrator who doens't have to do any deduction of give opinions of their own, at some level they're a clue-delivery mechanism.
With Watson things are different, he's a proper character with his own personality and opinions, but he's also the observer, the way we see the case.
Obviously he is smart, he's a doctor, he makes all the necessary observations to solve the case, because everything he observes we observe and everything he doesn't we can't observe, he does take over some of Holmes' deduction methods, but at the same time he can't be
too observant or too good at deduction.
If Watson too accurately notices certain clues we notice the right clues, if Watson deduces the solution then we deduce the solution.
Watson can never solve the mystery because then the reader has solved the mystery which means the story is over, there's no mystery story without a mystery.
Hence the incorrect assumption Watson can't be very smart, the man's a defacto detective for centuries and yet he never solves a case, but he's just cursed by narrative necessity.