@AvocadoInTheRain:
What did the theories of those people joining consist of beyond people just liking them and wanting them to join?
People in favor of Caeser thought he could reform, Perona supporters stated had a connection with Zoro and was one of the few people outside of the crew to know about the 3d2y message, Hancock's "love sickness" was played up in a way that many thought she would tag along. I don't agree with any of these, and obviously they stood no ground, but these are the arguments some people made that have while taking crew role out of the conversation for a second. Caeser has been continuously shown to be a piece of shit, Perona was simply serving story roles(being the first sent off by Kuma to make his appearance in TB impactful, and then to serve as an audience surrogate in the 3d2y plan), Hancock just didn't fit in terms of powers and a connection beyond Luffy(as compared to Jimbei with ties to Nami going back to East Blue).
None of them ever exhibited ship roles or dreams (that could be accomplished by joining the crew),
In terms of roles? You're wrong. Perona exhibited lookout abilities far beyond anything Carrot could do. In fact, the first time we see one of her ghosts, it's being used to scout ahead for her ship(Thriller Bark).
Dreams? Very shallow but Hancock joining could have worked toward her wanting to be with Luffy, given that she'll likely be his "chichi" by the end of the series. Hell, while he is in a happy position right now with his crew, many that people that I know that wanted Bartolomeo to join thought joining the crew itself would be his "dream" and that he would have been a spin on the typical "dream" aspect of the crew, with his being fulfilled and the rest of the story being him living his dream.
If it sounds like any of these are stretching aspects of the characters to fit them into the mold of a crew member, you're right. And that's exactly what is being done with Carrot right now.
I don't think there were ever actual theories proposed for these people joining, just a general sentiment of "I want him/her on the crew". Except for Perona, but that was literally one guy.
The things I just posted above are a few, but if you take the time to look back through the threads there are plenty of arguments thrown around for different candidates. Perona support wasn't coming from just "one guy". AGOG took the passion for his case a bit too far, but he wasn't the only person supporting it, just the most vocal.
As I stated, Perona would be the best lookout for the crew. She had a relationship with Zoro and Usopp(and briefly with Nami). Her ghost could probably serve a similar role comedy wise to what we have now with Nami and Zeus. Her power was shown to be unique and versatile. She had a very unique look for a girl in One Piece, if featured more she and her ghost would probably be heavily merchandised.
That being said, it didn't happen. As time went on she never had a dream or reason to travel with the Strawhats that was clearly laid out. Despite a unique design, ability that would serve a role for the crew in a unique way, and different levels of interaction with different members of the crew, she just wasn't meant to be a Strawhat.
People that were mentioned simply for "just wanting them" to join would be more along the lines of people that wanted someone like Keimi, who had no reason to travel with the Strawhats or a spelled out dream, yet people thought she would join the crew as their taylor and supply them clothing. Or people that thought Monet would join because she had a unique design, but no reason to travel with the Strawhats or a spelled out dream, yet people decided her astronomy studies must have been leading to a further explanation that would rationalize a dream to qualify her.
Honestly using every argument about Carrot outside of Pedro stuff can be substituted with Perona, except she'd actually standout as a lookout as opposed to simply filling it. Why is Carrot a "good" lookout. By "good" I simply mean "validating why they will fill this role that everyone else already fills". In terms of what Carrot can do in this position, it is all redundant. All of it. She is not the fastest, can't jump the highest, can't see the farthest.
In terms of designs compared to Perona, well she actually is unique. As others have stated in detail here, Carrot is not a unique character design wise because she falls into a general theme of a group of characters introduced at once. Everything about her appearance is simply "cute bunny girl" which would not be bad if not introduced at the same time as a dozen other minks that just look like stereotypical humanoid animals.
Can't believe I'm actually citing AGOG in 20fucking18 about a crewmate discussion. And yet for all intents and purposes with Carrot seeming more likely to join than Perona, the more I think about it Perona had more than Carrot to bring to the table and parts of the story to validate theorizing about her actually joining. The ONLY part that leads me to POSSIBLY believe Carrot is slightly ahead of Perona when looking at everything is Pedro.
If Pedro is alive, all hopes of Carrot joining are shattered and rendered moot.
If Pedro is indeed dead, then great, Carrot has a slight possible reason to join the Strawhats. But it's fully contingent on Wano being some haven of Carrot related story beats that lay out a dream for her, give her a reason to travel with the Strawhats specifically(Because as mentioned by everyone she has ZERO lookout feats that can't be done by others in the crew) as opposed to reviving the Nox Pirates, and Luffy to actually show even the slightest interest in her joining.