Does Sherlock really, "delete", things from his mind?
I don’t know if I believe Sherlock when he says he deletes things from his mind.
In the beginning of TGG we see John and Sherlock discuss his lack of basic astronomy knowledge. During the deduction at the art gallery, Sherlock saves the would-be bomb victim with some obscure astronomical knowledge that he seemed to dredge from pretty deep in his mind: if he’d just learnt that it might’ve come to mind more quickly.
We are not shown whether Sherlock spent all day boning up on his knowledge of astronomy or whether he already knew these things and was lying to John about it. We know that Sherlock lies; it’s one of the reasons why the original deduction of Mary as a liar did not alarm him in TEH. In TSOT Mary confirms that Sherlock not only lies but that she notices when he does so.
So, there’s a possibility that Sherlock attempts to delete things (like he attempts to delete feelings) but we don’t know that he necessarily succeeds nor do we know if he was telling the truth to John about the solar system. Maybe he was just trying to seem likeable like he discusses in MHR with regards to winking.
We see in ASIB that, "people want to know you're human", is John's rationale for putting an unsolved case on his blog. Sherlock has been constantly trying to impress John (and succeeding) since the moment he first met him. In fact, if the things he says he deletes are like his feelings then we know that he fails entirely. I see the solar system as symbolic of his emotions, something he claims he eschews as it has no purpose in his life. If it does help with the work (astronomy) or if it hinders the work (sentiment), he claims to delete it. But astronomy saves the child in TGG and sentiment turns out to be the real motivator behind him solving cases, anyway, like when he solves the Sholto case.
Further, the idea of the unreliable narrator is very strong in Acd stories. Moffat and Gatiss have gone as far as to say that when Watson claims that a mystery woman killed Milverton that they did not believe him. Being entirely too convenient an event considering the circumstances they interpret this version of events as a coverup of the likely truth: that Holmes or Watson killed him. Unreliable narration due to Watson idealizing Holmes is part of the relentless romantic subtext, as well: Watson would purposely obscure the unflattering or damning while also unknowingly over emphasizing the good due to his love and infatuation.
The idea that anything that the characters say could be false is heavily shown via various denials and admissions. Mrs Hudson's, "I'm not your housekeeper", (while constantly cleaning their flat and bringing food and tea) being one of the more didactic ones. We've seen that, despite his claims, Sherlock is not a sociopath. And we've seen that his Mind Palace contains pop culture information he claims to eschew in TGG. In THoB we see his MP contains such randomness as knowing Hounddog by Elvis, which seems like a likely candidate for his so-called deletion.
In HLV we see Sherlock's mind palace to contains forces he has attempted to put away such as Moriarty, representing his id. If he really were capable of deleting would he not have succeeded at discarding this feared part of himself? I think the whole point is that try as he might he can't. He can't delete anything. Not his emotions, or his sex drive or his ramdom bits of knowledge. In TEH, John suggests that maybe he, somewhere in his mind, knows how to diffuse a bomb. And though his first reaction is to mock the idea, he quickly concedes that, in fact, you never know what could be in there.
The idea that Sherlock can delete things from his Mind Palace I think then can be seen as possibly outright false: a way to appear more human to John by feigning ignorance. We see in TSoT that Sherlock begins the Rizla game with a joke and receives positive feedback ('am I a vegetable?'). He, soon after, mentions the current king of England. If we do not take this at face value we could see it as an continuation of Sherlock wanting to make john laugh; he succeeds. Or a continuation of him wanting to seem human to John. A kind of taylor-made modesty just for John. (We can see his, "I'm no hero", speech in TGG as possibly impressing John via genuine modesty).
We can also see his claim to delete things as aspirational: he attempts to keep his mind tidy of clutter, such as random facts and feelings, anything that doesn't help the work, but we see that he fails, at least with regard to the latter and that eventually both prove to be pivotal to the work, anyway.
