They do give a shit, Nekomamushi is rational type of character, he's been lied to by Carrot..
Also Emotional grudge isn't a free pass to justify every vendetta undertaking, can Carrot go down the rabbit hole and say she's going to kill Perospero and every one of his childs for forcing Pedro to kill himself? No
Her emotional grudge isn't warranted as she was told by Wanda AND by Pedro that this is not a game, the pirate world is no games..