@Chrona
1. The issue there is that it requires far more regulation, and thus power, to ensure the interior stays cold; the way you're describing it sounds more akin to an A/C unit than an ice box, which only power on when the thermostat registers that the ambient temperature is outside the set values. Which would work, but not in every climate. I'm from Texas, so my A/C often kicks on close to a dozen times a day during summer just to keep things at a near consistent 70F.
2. I haven't read very far yet, so I don't know if much more has been stated about her world, but how do you no they have no transportation issues? She mentioned a scroll of teleportation, but talked about it as if it were so valuable the only time she'd feel okay using it would be in a life-or-death situation. For the most part, it's far more likely that her world (as far as I know from where I am in the story) is the typical fantasy medieval setting with a dash of magic, so the majority of things are done the same way they would have been in the 14th or so century here on Earth (with a few exceptions).
I don't doubt that they've come up with the concept of an ice box using magic, because it
is a simple concept, just that they seemingly do not have the ability to implement it reliably.