the entire thing with Naoto and Takemichi is that Naoto's desire to save Hinata sets off the time travel handshake.  Once she's safe in the future, he no longer has that desire and the handshake no longer works.  Takemichi dies while saving Mikey from jumping off a building, but in grabbing his hand and declaring he wants to save Mikey, that sets off a desire in Mikey to be saved, and the handshake triggers.  Meaning it's not special to Naoto.  It also means that Emma's murder was even MORE irrelevant, because by this point Mikey knows about Takemichi's time travel ability and that he could absolutely go back in time and save her, but instead wants to be saved himself, so the handshake between Mikey and Takemichi sends Takemichi back to long after Emma's already been killed.  So I guess he never really gave a shit about his sister, and she was absolutely, 100% entirely kept around just to be fridged to motivate the male characters.