as one of the most reasonable Carrot fans, I’m really curious what are your thoughts about these 3 main points to be a Straw Hat:
1. Every single Straw Hat was a co-protagonist who needed the help of the current Straw Hats to finish their personal conflict
2. Every single Straw Hat has their own main villain of the arc, whom in the end got defeated by Luffy with the help of Straw Hats; as the main villain’s underlings got defeated by current Straw Hats / the potential Straw Hat
3. Every single Straw Hat has terrible past which makes them more like a family
let’s admit with the fact that Carrot fulfills none of the points above. Care to elaborate from your POV why Carrot is the only one special to be a Straw Hat without fulfilling any of main points above? Because no Carrot fans can explain so far with proper evidence & facts, but headcanons.
Alright, had a fresh night's sleep so now I'll take a proper crack at this question. I think the issue that I was having yesterday is that I was trying too hard to find a way to stretch your first two points over Wano in some way. I can explain how she fulfills all 3 in Whole Cake Island, but she does not fit into the first two in Wano at all, which I'll get more into later. So instead of trying to force a negative into a positive, I'll instead acknowledge the negative and bring up how she might make up for this later.
I would say that Carrot just misses fulfilling the first two in Whole Cake Island due to its unique structure. Most other arcs have Luffy beating the big bad, saving the island and its people with the help of his crewmates knocking out the upper tiers in the villain's group. The potential crewmate officially joins when the dust starts to settle. However in WCI this does not happen, Luffy doesn't beat Big Mom and save WCI's citizens, no one in the arc has a major 1v1 other than Luffy, and the arc ends with the crew trying to flee from the BMI pirates.
Then the story goes into the Reverie arc straight away, which then instantly leads to the Wano Arc...
Whole Cake Island ends at 902, Reverie starts at 903 and ends at 908, Wano starts at 909.
If WCI was a standard arc then Carrot would fulfill all 3 requirements easily:
1. Personal Conflict with Big Mom and the BMI Pirates and how they kidnapped one of her saviors that saved her people, who just so happened to be a Straw Hat.
2. Since BIg Mom is the mastermind behind the kidnapping then she would be Carrot's main villain that Luffy would takedown, with her and the other members taking out the upper tiers of the BMI pirates. During the after-party she joins and no one bats and eye
The third one is one I think she still consistently has today and that's with the death of Pedro. I'm in the camp that Carrot is an example of a real-time tragedy. Arguably, the whole scene of Pedro sacrificing himself while Carrot tearfully and powerlessly watches could very easily be compared to the flashbacks of other Straw Hats. You mentioned how the terrible past makes them more like family and I would agree. I would argue that through the tragedy of Pedro's death Carrot receives a heartfelt scene with Sanji which leads into the family moment that you mentioned. You'll see both pages that I'm talking about linked below. That moment where she's cheering for food with the rest of the crew helps to tie a neat bow on the end of the WCI arc.
I swear to god though if Pedro is still kicking then that'll be some real shit, wouldn't it...
Anyway, since the first two didn't actually happen then that means Carrot still needs to finish these requirements off in the current story.
The issue is that now Big Mom might be beaten by someone other than Luffy and the Straw Hats, not only that but the BMI pirates are currently out of the picture. There's a chance that Oda might goof them out of the arc entirely instead of them being Carrot's personal conflict by trying to kill her saviors again.
So the only real chance of Carrot completely tying the loop on WCI is that Big Mom and the BMI pirates escape Wano and become the villains of another arc. But let's be honest, absolutely NO ONE would be a fan of it as I think we're all ready for some new villains in the story.
And the elephant in the room here is that Carrot does not fulfill these first two requirements in a satisfying way in Wano. I'm pretty sure that she has not shown outward emotional investment towards Kaido and the Beast Pirates. And while the rest of the crew have been fighting the Beast Pirates, Carrot's only fight was with Perospero which she lost.
This is actually pretty crazy if you think about it...
even now she's been focused on WCI related plot points with Perospero.
So to put in words what this means, I would say that with the narrative side and the friendship side of being a nakama Carrot is much weaker on the narrative side compared to other Straw Hats. The other members that took more than 1 arc to join (Nami, Robin, Franky, and Jinbe) had strong Straw Hat vibes in each arc. Unless Carrot's situation changes then she won't have that luxury considering her current status in Wano.
However, I do think that Carrot can potentially make up for her weaker narrative side with her stronger friendship side. I think there's a legit argument that Carrot is a true close friend to the Straw Hats, which could potentially make up for her shortcomings in Wano
But again her current situation would need to change first before that can be properly explored...