Sorry about the length and untidiness of this post. I've tried to cover many aspects and give my full opinion on this topic as it matters to me quite a bit. To understand this topic, you should already have played through the story, so I will not be marking spoilers.
I strongly sympathize with people who have been hurting by the unfair treatment of their favorite squid. It's to be expected for some rival fans to twist the knife, but when the developers themselves pile on (inadvertently or not), it's a helpless and despairing feeling. I feel like that is the root cause of the movement and why it has effected such a strong, seemingly indignant reaction that may puzzle others.
Consider that Callie fans have had to deal with losing THE splatfest, which in my opinion only ultimately served to foster bitterness and resentment between Callie and Marie fans instead of resulting in anything positive, but also that there's no sign of ever being able to "redeem" that loss. It's not like a sports team losing a big game and being able to find solace in "there's always next season." It's something that will always sting to those that care. Then seemingly right after that, the events of Splatoon 2 happened as an additional punch to the gut. The old cheerful, ditsy Callie that her fans grew to love was gone. It can be argued that Callie was gone altogether since she literally wasn't herself aside from a few moments when she was trying to warn Marie to stay away. She had been turned into a villain and a robotic shadow of her former self.
I also think they went too far with the outfit, especially with the tattoo. Though it's clearly only meant for stylistic purposes; it also aids in the mudslinging against her and it's hurtful every time I see it because Callie doesn't deserve that sort of treatment or connotations towards her at all.
I take issue with writing her to insist on keeping the glasses. Some people take that to mean that she must enjoy being brainwashed even though the glasses certainly would have no hypnotizing effect after Octavio is recaptured and the Zapfish returned; they would only function as a cosmetic item, which is what she explicitly states they're for. The thing is, that sequence only happens after you replay the boss fight and Callie is still there, still brainwashed, so it conflicts with the idea that the glasses are now only a cosmetic item even though the whole thing is clearly meant as a comedy bit and to offer some sort of explanation as to why you can just go and replay the boss fight and that's all at the expense of following consistent logic. Octavio must re-escape from the snow globe every time as well only to be put back again, which makes just as little sense, but that doesn't stop people from taking the retainment of the hypnoshades at face value. It's just one more little thing on top of it all. It can't help but cross my mind that the writers are intentionally mocking and making a comedic pinata out of Callie at this point. Callie has always been the butt of jokes due to her naive nature, but those jokes were innocent enough and importantly they were always delivered by Marie. This is different and the whole thing has taken on unsavory tones that I don't care for at all.
One thing I really am curious about is what fans in Japan (who are perhaps more connected to the developers) think about all this and if they have a similar subset of the community that's unhappy with how Callie was treated and depicted in this game. I've heard before that Callie is a somewhat different character in the original Japanese localization and perhaps it isn't as jarring to them, so perhaps the developers wouldn't see any immediate problem with it. Just speculating there, but it may be a very important point and help to explain their apparent obliviousness if true.
As others have pointed out already, they made a prologue story to the game that involved Callie and Marie growing apart after the final splatfest and briefly reuniting before Callie was abducted at the beginning of Splatoon 2. This was not paid off in the events of the game. It was not paid off in the ending or second ending either. Does it bother Callie fans that she was abducted and brainwashed (by a giant evil octopus of all things) only to be ultimately barely in the game at all? Of course, but at the end of the day all they're asking for is some closure on this uncomfortable and distressing affair and they can't even get that. All it would take is for Callie and Marie to visibly reunite and just hang out somewhere in the Square, so as to say that everything is back to normal. If the developers really wanted to apply themselves they could even have them rotate in for Pearl and Marina during splatfests (if you've beaten the single player).
As an aside, I was always inclined towards Callie, but I was never a die-hard supporter until after she lost the splatfest and after I had seen her unfairly drug through the mud and continue to be drug through the mud by so many and after I had seen what she was forced to become in Splatoon 2. I think it to be tragic considering Callie's core character as being innocent, cheerful and naive for something as unfortunate as this to happen to her. I feel the same kind of sympathy that I'd have for a child that has something bad happen to them and not only did that bad thing happen to an innocent, but because of that innocence, they can't even understand why this thing has happened to them. I realize that sounds like an extreme comparison, but note that I said the same kind of sympathy, not the same degree.
I don't hold out any hope for this movement to affect any change in the game itself even though the changes required would be slight. The way I see it, if people want to draw pictures of Callie and show their support for her even in the face of such neglect and indifference by the developers themselves and use it as a sort of coping mechanism during the long wait for Splatoon 3 (which let's be honest is most likely the soonest point we can realistically expect a resolution) then I fully support it and appreciate their efforts. Thank you for hearing me out. Long live Callie.