I'll start by saying that I absolutely do not agree with the notion C4N brings up that "they're not needed" since as you say it is something that can apply to all the other Straw Hats, which includes Vivi and even Yamato. Especially since one could argue how Carrot does fit into these points which could potentially help her position as a potential crewmate. The glaring issue here is that as you said, one needs to explain why Carrot is special when compared to the other Straw Hats.
I may have a satisfying answer to this, using actual evidence from the manga as well. What I did was that I reread Zou to look for clues and see if there was any confirmation that WCI was always meant to be an abnormal arc. In my mind, to prove that WCI is truly unique could help justify why Carrot is unique compared to other Straw Hats.
Here's a breakdown of how WCI is unique, the first 8 pages are from Chapters 814-815. The last page is from Chapter 819. After you read through these pages I will give a page-by-page breakdown of how they prove WCI's uniqueness. Note that in all but the last page Carrot is right there with them. After the breakdown, I will try to explain why Carrot is a special case Straw Hat.
- We get an explanation from Zoro that the threat of Kaido and the Beast Pirates is now a reality that can't be ignored. The timing of Sanji being kidnapped was horrible as the crew was already on an ultra-important campaign against one Yonko.
- Extension of the first page, Zoro explains how they're already on a collision course towards Kaido. After Luffy suggests simply going straight to Sanji he is reminded by his crew that Big Mom is a force to be reckoned with. Most importantly, we see Robin saying that "She's an Yonko, Luffy!! Don't forget this isn't like before." Luffy then responds that they'll "go quietly." I would say that this confirms straight away that Big Mom is not like other arc villains, even Luffy doesn't want to simply tackle her straight on.
- Pages 3-6 are an absolute dump of info that explains Sanji's situation and Big Mom's massive influence and power. Not only that but introducing more potential danger in the situation with Germa 66. Oda spends a lot of time making sure that the audience knows that Big Mom is in a completely different league when compared to past villains, even ones like Doflamingo, Crocodile, and even CP9.
- The important part on page 7 is the beginning of the Sanji Retrieval Team. Luffy initially declares that he wants to go alone, with the threat of Kaido they can't risk an all-out war with another Yonko. Luffy actively wants to avoid fighting and just wants to focus on saving Sanji. Also, there's a neat little seed planted here with Carrot saying out loud that a rescue mission sounds fun.
- Page 8 is a reconfirmation that the Straw Hats can not simply ignore Kaido and focus on saving Sanji.
- The most important part of this page is that Luffy declares that he is not fighting Big Mom.
So putting all of this together, I would say that it proves WCI's uniqueness in 3 ways.
1. The scale of Big Mom's influence and power is one that we've never seen in the series before with other villains. Facing BMI head-on is not like the raids on an Alabasta city, the Dressrosa Kingdom, or the CP9 HQ. It would be an all-out war against a full-on country.
2. This is a rare scenario where the Straw Hats were facing multiple strong forces at once, the only other time that I remember was during Thriller Bark when they faced 2 warlords back to back. That scenario is a perfect example of how dangerous a 2-front battle is as well, any other Warlord other than Kuma would've wiped out the Straw Hats right then and there.
3. This is another rare scenario of when the Straw Hats wanted to actively avoid fighting. The last time was during Alabasta when they were originally going to convince the rebel leader to stop the attack on the kingdom. But plans changed and they went to defeat Crocodile instead. What makes WCI different is that when the plan changed with Sanji's rescue they had to run away instead and escape WCI after a failed assassination attempt instead of trying to defeat Big Mom and her crew.
This brings back those 3 Straw Hat points you mentioned.
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, who 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 a terrible past which makes them more like a family
I feel like I can confidently give true answers to each of these now, the main thing to realize is that with WCI's uniqueness that each of them has a slight twist.
1. The twist here is that the Straw Hats are the ones that need help this time instead of vice versa. Carrot fills the co-protagonist role by joining the Sanji Retrieval Team and consistently aiding the Straw Hats in their personal conflict to save Sanji from Big Mom. The pages I presented earlier show that Carrot was interested in this rescue mission early on in its development.
2. Obviously, Big Mom is the main villain of the arc. The twist here is that she didn't get defeated in a traditional way. Below are two pages from chapter 903.
While some of the information is a bit misleading, Luffy was actually victorious due to saving Sanji while beating a couple of top-tier officers and destroying BMI's main castle. All with the help of the co-protagonists this arc, which includes Carrot.
3. I've talked about this before in my previous response to you. Basically, the twist is that Carrot's terrible past takes place in real-time, watching her mentor and one of her closest friends die right in front of her eyes.
So now it's time for the golden question, why is Carrot so special compared to other Straw Hats? This one I had a lot of trouble answering, but after rereading Zou I believe I have a legit answer.
What I present is that if Oda truly intends to have Carrot join the crew then the only chance he had to properly lay the foundation for her being a potential crewmember was during WCI. But since WCI is a very unique arc, Oda wasn't able to lay her foundation in a completely traditional way.
First I would like to go back to my reread of Zou, something special about that arc it sets up 2 arcs at the same time with WCI and Wano. Keep in mind that during the talks and set-up for WCI you can see Carrot right there with them, which you can see in the pages I linked above. But during Chapters 818 and 819 the official talks and set-up for Wano in Raizo's cave Carrot is nowhere to be seen.
The most important part of that last page is Carrot saying that she wasn't even born yet when it came to the tale of the insanely important connection between the two Mink Kings and the Kozuki clan. Combining her lack of presence during the scene in Raizo's cave + the fact that she didn't even exist during Oden's adventure shows a potentially interesting revelation.
This revelation is that she has almost 0 personal conflicts or connections to Wano and the Beast Pirates. Instead, the person that does have all of these personal connections to Wano and the Beast Pirates is Yamato.
This is more of a personal theory, but without her involvement in WCI then it would be completely impossible for Carrot to join during Wano. She would have to lay a traditional foundation with your 3 main points alongside all of the emotional explorations with Wano's people, Oden, and the red scabbards. As well as with Yamato laying her traditional foundation. And all of the personal moments with the other Straw Hats. It would spread out too much focus in an arc that's already criticized for being spread too thin.
And to have her lay the foundation in an arc after Wano would be almost impossible as well. Due to the fact that if she joined after Wano without the foundation of WCI then it would be like a complete stranger with almost 0 connection to the crew joined out of nowhere. Even if she joined right there it would still be hard to lay the traditional foundation as there is a good chance that the Straw Hats go straight to Vivi/Sabo instead of dealing with any personal conflicts that Carrot would potentially have.
This leads to where we are now today with Carrot's status post-1006. Which is waiting for an emotional redemption moment that would potentially use the Wano Wildcards (BMI Pirates and the WG).
I spent way too much time on this, I'm going to go pass out and sleep now...