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cor3zone
Sure, Kaido is capable of acting alone, but this doesn't mean that S-Ranks are "expected" to do so, but rather that at least some of them are "capable" of doing so. (The rules of the Guild are for all ranks, as evidenced by the fact that Kaido refers Carol to a rule she was aware of, back before she knew he was S-Rank, which shows it applied to other ranks, and, with only 10 S-Ranks, those rules would be written with normal ranks, not S-Rank primarily in mind, so it tells us nothing about S-Ranks being treated in a special way.) Kaido is likely special, even among S-Ranks, because he has skills is swords, offensive magic, limited healing magic, and uses spirits. (I wouldn't be surprised if it is revealed that he is one of the princes of the kingdom, and that royalty get special blessings.)
The evidence we have is that Kaido doesn't normally pair with people, because he finds it a pain. He details his reasoning. Since his boss mentions how stubborn he is, there is ample evidence to believe that he insists on working his own way, with his own team, regardless of what they say. (This includes the fact that he continues to insist that he be in a team with Carol, even when told not to team up with her, showing that, regardless of their desires for the composition of his team, he still decides how to team up, himself. This evidence, directly from the manga, counters the argument of more evidence of expectation to act alone, because he is shown to go directly against the "expectation" for what his party should look like.)
Yes, the magician underestimated her, but not anymore that your post about it shows that you are underestimating her. He is still extremely smart, and had been doing research on her for more than the three days she had to do research on him. As such, his preparation should have been good, too. He thought he knew what she was capable of, and acted accordingly. Again, the thing he claims to be best at (knowledge) is the very thing she beat him at in their contest. To see it as anything other than a display of tremendous competence is to be intentionally blind to her accomplishment, just to undermine her capabilities for an argument. There are plenty of people that are far from "complete morons" that would fail. She had to design an entirely new product (and did so within 3 days) in order to accomplish it, but she did so.
You say that because they are "emergency requests" that Carol wouldn't have time to prepare, but that is proven not to be the case in the manga. Despite them having to leave right away, she is still able to concoct a large amount of Mountain Ape repellent, which shows itself to be highly effective. Sure, she couldn't hunt them down, but the request was only to investigate, not hunt them down. She is actually the one that discovered what was going on (that the new type was a Boss Mountain Ape), and how to solve the issue. Thus she actually did more of the actual quest completion than Kaido did. Kaido just helped to take out the monster, after discovering what it was. (In many settings there is are "investigation" quests that are quite different from "subjugation" quests.) Not to mention, that even emergency quests require time for travel, which also gives her time to prepare. It would be silly to think that they send people in without letting them do what is necessary to prepare.
If they get another emergency request, then they either send somebody else, she quickly replaces her stock and moves on, or they wait a couple days. To reverse things, what do they do if Kaido's weapon is damaged by one quest? Do you think they say, "It's an emergency so go fight without any weapons and hope your magic is enough, but if you die, whoops"? (Of note, Kaido is shown needing to repair his sword, and that the repairs will "take some time" so what if he got a request during that time? Does he rush off without a weapon because it is an "emergency"?) Proper equipment is also a part of an adventurer's life and is factored in on which quests to take.
Yes, I am aware of Achilles, and his heel wasn't "cut" but hit with a poison arrow. Kaido was close to death, and his efforts of fighting the worm were actually making the situation much worse, according to Carol. He had actually accelerated his death, rather than fighting it off for longer. It wasn't the length of time that had made it grow, but his constant attempts to counteract it by healing himself, which made it get more food. So, rather than showing his competence in that situation, it actually showed that he had done the exact wrong thing, and that he is lucky Carol was there and knew how to treat it.
He has other shortcomings, too. In the fight with the Inozasaou, he was capable to defeat it, but not to save the lives of the villagers. Both Carol's healing, and her ability to attract it away from the building they were in, were necessary for those people to live (as Kaido again acknowledges). Her "search" is also bigger. As for her inability to fight, I question if it will always remain such, or if part of that was the fact that the party she was in was focused on continuing to use her and make her feel useless. They used her strength to boost their own ranks. Yes, she doesn't know the "formal" name of the kingdom she is in (which is why it could be possible that she is traveling with a prince and not know it), but it isn't like that information is necessary for adventuring, or would, in any way, stop her from accomplishing a quest. Have you ever read a quest that said: "Come to x village and teach them the formal name of the kingdom, rather than the name it is generally called. P. S. This is definitely an emergency S-Rank quest"? Of course not. The idea that knowing that is a barrier to S-Rank is silly.
Those are also questions that are common for those living in the capital city, but would be less known in more remote villages, where she is from. If the question were to name all of the various families in her village, and the rough population numbers for the past 5 years, I bet she would ace that, but somebody like Kaido would have no clue. "General knowledge" differs based on region, and she comes from a further region than many. It is interesting though, how quick you are to dismiss the "specialized knowledge" that is actually "knowledge beneficial for a healer to have, and the more they have the better they are at healing". Funny that you want to focus on the part that, as you say, it "a moot point" because it doesn't actually affect her rankings, but in doing so also try to treat the knowledge that should affect her rankings as a "moot point" as well.
I like how, in one paragraph you try to say that Kaido's competence is such that he doesn't need anybody, then in another paragraph say that his incompetence is actually Carol being a hindrance to him. So he's S-Rank when he's alone, but drops down to B-Rank when she is with him? That doesn't make sense. You are bending over so far backward to dismiss her abilities and competence, that you would rather interpret it as Kaido being lazy, than that she was able to do something he couldn't, and saved her life? Seriously? Do you just hate her or something? Also, he "specifically requested her help" because he knew that she had skills, abilities, and knowledge that he didn't. He normally preferred to work alone, so him wanting her to join him is not evidence of him becoming lazy, but of just how excellent she is at what she does. To interpret it other than a declaration of her strength is to ignore the will of the author.
As for you question, "Could Kaido?" The answer is not "Possibly" but "No" as declared by Kaido himself. It was Carol that detected the Mountain Apes, it was Carol that discovered a leader, and it was Carol that told Kaido what to do to fix things. The manga tells us that Kaido couldn't do it on his own, and the only reason to interpret it otherwise is to minimize Carol's accomplishments. Again, you sound just like the court magician, who said Kaido might be over-exaggerating her contributions, which greatly offended Kaido, because he was adamant that he was always truthful. Since you have shown your thoughts on the court magician (using his own talking points), I would recommend not sounding like him, if you don't want to be in the same category he is. Kaido is offended by those who question his integrity, and clearly has a strong sense of justice. As such, there is much more evidence to suggest that he would have died at the hands of the Mountain Apes (albeit, while killing many of them) than that he could have soloed the encounter if Carol wasn't there.
Parties combine "combat-ready adventurers together with non-combatants" when they form parties to begin with, so that part of the argument doesn't make any sense. If they evaluated it otherwise, it would be difficult for "combat-ready adventurers" because they might take damage, and then have to return and heal, before moving on again. Or you'd have to assign them jobs that would allow them to never get injured in the first place, which is difficult to do. The fact is, fighters are much less effective without a good healer. Look at Gran's group, for example. They looked like an "A-Rank" party, when they were a bunch of scrubs, because she kept them healed. It would be hard for Kaido to keep a party that incompetent alive when facing A-Rank monsters, but Carol could do so. Dragging a party of people that weak far enough along for them to get A and B-Rankings shows just how strong she really is. A B-Ranker that specialized in healing was not even close to strong enough to overcome their shortcomings, but Carol was able to do so without issue.
Even in combat, there are different types of quests. This enemy is immune/resistant to magic. That enemy has an incredibly strong physical resistance, but is susceptible to magic. The fact that they have handlers already shows that they are making evaluations based on where people are and what abilities they have for quests. They have 10 S-Ranks, currently, and it is highly likely that they already determine which S-Rank fits the bill for certain requests. All requests are also prone to mistakes. There are many manga that have stories of an "E-Rank" or "D-Rank" quest that was mislabeled, and if not for the MC being stronger than evaluated, the party would have been wiped out (or other parties do get wiped out). The fact that they have different ranks for different quests already shows that they have to evaluate quests and figure out what is needed for them. As such, what you seem to treat as more complex is actually what they were already doing. Also, in your example, the "fix sickness" quest would have been put in the "X-Rank" category, and you still end up in the same situation, don't you? How does that stop the mislabeling from happening in any way? It doesn't. Adventurers are responsible for paying attention to which quests they take, and making sure they are capable of completing them. Handlers do their best to recommend quests that they think the person could complete, but they make mistakes, too.
To go down the rabbit hole of "S-Rank" and "X-Rank" quests, then you need to have different ranks for "Fighters" "Mages" "Magical Swordsmen" "Spirit Users" and "Healers", meaning you now need 5 different letters. Oh, and we also need letters for combinations of those groups. What letter do you use for a quest that is "Unclear" and needs both? "V-Rank"? (Also, the idea of you suggesting sending both is confusing to me. You said that "S-Rank" are "able to act alone" so wouldn't sending an "X-Rank" mean they weren't really an "S-Rank" to begin with, since that means they weren't able to "act alone" but needed an "X-Rank" with them? "S-Rank" means they can handle everything, right? Thank you for showing that there would be expectations for them to act in groups, even after saying there wasn't evidence for it.) Adding more letters actually makes it more likely that a quest gets mislabeled, and more likely that accidents happen as a result. Better to have one letter for the highest of all classes, and let the adventurers pick based on what abilities they have, and expect that specific requests to one of those individuals is based on their specific abilities (because if they aren't, anyway, whoever is making that specific request is stupid).