Moving on to the topic of Carol Holiday in particular, there's a specific trait of hers that I think deserves a lot more attention and analysis: her refusal to keep anything of importance in her office.
We know the reason for this - it's because she thinks it's in danger of being damaged. The question is, why? It's Carol's office - and she spends more time there than she does in her own home! Yet she refused to put up the snowflakes Noelle made, even after she had them bronzed. She won't even keep a spare key to her house there, despite all the headache it would save.
And the thing is, her house isn't actually that much more secure than her office. Even when she's not there, surely the door is locked! And it's not like the fence makes that much of a difference; neither Kris nor the Wet-Nose Bandits were deterred by it. Which tells us, at the very least, that this behavior is anything but rational - in fact, it's outright paranoid.
It's not hard to see why she would act that way, in the abstract - Carol already lost one daughter, and she clearly doesn't want to lose another. But the way this behavior plays out is very strange. Most overprotective parents act more like Rudy - going out of their way to shield the kid from perceived threats or conflict, to the point where their child never develops the ability to stand up for themselves. Yet Carol is, instead, incredibly distant - seeming only to interact directly with Noelle when there's some kind of problem.
Carol reminds me a great deal of my father, and I'm going to have to offer a bit of context as to why.
My father became distant when I was very young. It used to be that we'd hang out all the time - I'd follow him to wherever he was working on a project for his dad, and enjoy looking at all the machinery and tools while he did his thing. But when I was little, that stopped suddenly.
It wasn't malicious: he was trying to protect me from his father, an evil, paranoid man who abused his older children and freely accused people of stealing his possessions any time he lost track of them. My dad has severe C-PTSD from growing up with this man, and spent much of his life terrified that one of his children would stumble into the wrong situation and unwittingly cause a catastrophe - not just with his dad, but with any potentially unsafe situation.
And yet, it was him, and his unmanaged anger issues from that trauma, that posed one of the biggest threats to us growing up. And he knew that. He knew the damage his father's anger had done to him, and that was another reason that he withdrew from his children - he didn't want to hurt us the way that he had been hurt.
And that is very much how I read Carol - as someone who is not just afraid of outside threats, but afraid of herself, to some degree. This is the same woman who hired Asgore to clean her home, despite seemingly not trusting him to protect the shelter code. And yet she's afraid of - what? That someone is going to burgle the mayor's office? When everyone knows that she keeps a fucking sword on her person and is fully willing to use it? The only meaningful threat to the contents of her office is Carol herself - and I think she's self-aware enough to know that.
I'm not saying that something happened to Carol in her childhood. Obviously that's unknowable, and irrelevant to boot. What we do know is that, although she has always been a somewhat cold and distant parent, these problems grew even worse after Dess disappeared. And that tells us one important thing: Carol is afraid.
What's she afraid of? Well, a lot of things, probably. No doubt she's trying to protect Noelle from whatever happened to Dess - something that, considering her likely involvement in that event, she probably feels like she was directly responsible for. And she's probably just as scared that she'll never get Dess back - a goal that she seems to be working toward, and is no doubt terrified of fucking up.
From there, I think it's easy to understand a few other things about Carol. Notably, and very importantly, she is incredibly risk averse. Whether it's about her children's and valuables' safety, or being able to trust other people and ask for help, she's not the type to take even perceived chances.
This doesn't seem to be out of any hatred or malice - the opposite, in fact. Carol seems to be someone that loves and cares deeply, to a point where any kind of loss or betrayal would be outright devastating. And so instead of letting herself actually experience or enjoy her relationships, she tries to lock them up in glass cabinets, where she can admire them without risking any possible damage.
But the issue with Carol's approach - and the reason why this makes her such a terrible parent - is that relationships aren't objects. They're living things - they need to be tended and cared for to stay healthy. Freezing the world in a climate of ice doesn't preserve its contents; it just prolongs the death of winter, with no spring in sight to regrow what's being lost.
When we frame it in this way, then Carol's relationship to Deltarune's overall themes - stagnation, avoidance, and the need to face your fears before they consume you - becomes clear. Carol's fatal flaw isn't really refusing to let Dess go: it's trapping the idea of her in a state of perpetual teenhood, and refusing to engage with whoever she might actually be now.