Since the maxim is initially delivered by a detective who is trying to solve human crimes in a realistic universe, I think we can safely assume that by "impossible", he did not mean to descend into solipsism.

Otherwise every locked-room mystery could be solved by "the killer traveled through a folded pocket universe", or "the killer had telekinetic powers and managed to stab the victim in the back from outside the room". And similarly, seances and faeries are improbable enough to be grouped with the impossible for now.

Me quantum-tunneling out of my chair to Mars is improbable, but so improbable that it can be considered impossible... an event that will certainly never occur within the entire lifetime of the universe is indistinguishable from one that cannot occur. At that point, differentiating between the improbable and impossible is splitting hairs.

But at least quantum mechanics exists within the framework of what we know to be provable reality. The supernatural doesn't - making it even less probably than me spontaneously teleporting to Mars like John Carter.

The spiritualism that Arthur Conan Doyle believed in and the seances he attended were fraudulent, as were the Cottingley Faerie photos (which he believed were real because the photographs looked convincingly real to him.) I'm certain that he eliminated all other mundane possibilities from his mind to arrive at the supernatural conclusions he did, but he was still wrong.

Mere process of elimination is not sufficient to prove something "however improbable." Once you eliminate the impossible, whatever remains, however improbable, still needs to be proven. Otherwise you wind up in the trap of confirmation bias.

Yes. Conan Doyle's maxim is apt - it's just he misapplied it. In his consideration of the case of Cottingley Fairies, he wrongly considered the girls as too unsophisticated to be able to pull off a deception, and too honest to want to. He eliminated the wrong elements.

That's the problem - the maxim only works in a fictional universe with a protagonist like Holmes who has an uncanny ability to deduce elements of the plot with perfect accuracy from details as minor as the angle of someone's shoulders or the way they tie their tie.

It doesn't work in the real world because it first requires perfect foreknowledge of all of the possible explanations for a phenomenon, as well as perfect confidence that the attempts to disprove all but one of those explanations were correct.

But, as demonstrated with Arthur Conan Doyle and the Cottingley faeries, there may be assumptions one is not willing to challenge (the literal existence of faeries themselves) and possibilities one may not have considered.