What's the point of having a backstory when your tragic event is happening in real time. It has the same impact as a backstory.
luffy "sad backstory" was Ace lol. Who told you that these were the criteria for joining the crew. The only thing that just the SH have in common compared to the other is their role in the crew and Oda confirmed it. Oda never confirmed that he recruited the members according to the criteria you mentioned. Otherwise all the princesses are SH
all the set up of Carrot is done on WCI (role on the ship, tragic event, inherited will, crew dynamic with the SH) Wano was not her arc and it doesn't erase WCI.
A lookout is so important, it allow everyone to simply fit in their roles without having to fill multiple position.
the SH didn't need a lookout why did they need Bonney's help to make them notice the danger.
Many characters are linked to the SH but don't have a crew dynamic with them like Carrot. Carrot actively participates on the ship and even has a role on the ship while traveling with them and warns the helmsman and navigator of the threats. No fellow travelers have had that
"swears to protect Momo even if she has to die for it" is a fact
"tells Kaido that she can't be called Oden if she doesn't defend the country" is a fact
"having the DF of the guardian protector of Wano" is a fact Oda can't have introduced a DF with a literal name like that if it's not to develop it later on it would be a waste
says she was going to stay in her country and asks them if she is one of them exactly like Vivi did in Alabasta that's also a fact
dawgg "tragic event" Carrot has a tragic event to which she received the will of her mentor as all SH