Hi~ I just rewatched TGG, and was curious about something: what do you think motivate Sherlock to have John deduce the shoes? This immediately follows John scolding him for how he treated Molly, and it's always struck me as a bit...odd, that he jumps from "what do you mean I was rude" to "well go on then, deduce me some sneakers".
To me, the sudden change in topic is key to understanding what’s going on here. Throughout the episode, John and Sherlock disagree about their fundamental motivations. Sherlock praises Moriarty for being interesting, he is apparently more upset about being outplayed than people dying, and he believes approaching the situation with detached logic is the best method.
John meanwhile, is far more concerned with the lives at stake than beating Moriarty at his twisted game. He knows caring won’t help save the victims, but to him that empathy is extremely important.
This is introduced subtly in the beginning of the episode during the conversation about John’s blog. To Sherlock, the facts he deletes are irrelevant, while to John they are practical things everyone should know. The entire episode is about the two sort of overcoming that difference in opinion, and in the moments between solving crimes it’s mainly what the two leads talk about.
So back to the scene with the shoes. Sherlock, trying to help Molly, coldly informs her that her love interest is gay. John berates him for this, telling Sherlock that what he did wasn’t actually kindness. Sherlock then tries to show John up. Sort of like “Well, maybe I can’t be tactful, but I can solve this puzzle and save the person wrapped in semtex, and I know you can’t.” So he talks John into playing along. John instantly knows what Sherlock is up to “I’m not going to stand here and try to disseminate a shoe so you can humiliate me.”
He does go along with it though and Sherlock is allowed to prove his point so the conflict can continue to build. It only fully resolves during the pool scene, when Sherlock reveals that he does actually care in spite of himself. This inspires John to sacrifice himself to save Sherlock, and in turn for Sherlock to compliment John on his bravery and loyalty. It’s important turning point in their relationship, and one that they needed to get out of the way early on.