What if Addison lived? Because I will do anything but finish that fic in my WIPs
Prepare for a LOT of speculation.
For whatever reason, Larson's plan to sacrifice his daughter goes awry. He does something wrong and Addison lives, but it is excruciatingly clear what his intention was, even to his young daughter. As a result, Addison never trusts her father again. The attempted sacrifice drives a rift between them that will never go back to normal. She makes a point to stay out of his sight and won't often stay alone in the same room with him, growing skittish and cold. After all, part of her is afraid that he'll get it in his head to try it again (and he very well might!). If Uncle is present during this time, she would probably stay with him more often than not; the only family that she has left, and at this point, the only family she feels like she can trust. Addison would grow up desperate to escape this town that bears no name (because with no sacrifice, Larson does not name it after her), but there are next to no ways out that don't involve Larson somehow. The town has eyes and ears that all go back to him. Similarly, the mines remain open since there is no monster within them, and nobody is any wiser.
Until she can get out somehow, she spends much of her time in the town: getting into all kinds of trouble, learning from the miners how to shoot and hunt and camp and do all sorts of unladylike things that Larson hates. Nothing he does to keep her under control works: scolding her? She laughs in his face, or worse, goes straight for the throat with the unspeakable deed he tried to commit against her only a few years before. No dinner when she's out too late? She sneaks down to the kitchen after he's asleep. He locks her in her room? Someone taught her how to pick locks. She is impossible to argue with and impossible to contain.
Sometimes she stays away for days, camping out in the woods even though they're stalked by wolves. Some of the miners, despite themselves, have taken to keeping an eye out for her, making sure she doesn't get herself into something she can't get out of. People who come to visit are likely to hear the whispers about that wild Larson girl that tarnishes her father's good name; the one who wears borrowed denim and wears her auburn hair loose and tangled and laughs loudly at the crass jokes from the men in the Red Right Hand. She's rough around the edges and devastatingly vibrant, unlike her refined, reclusive relatives that rarely come down the mountain at all. She has a look in her eyes that makes them take pause; a look that is so much older than the girl who wears it.
Option B: Larson Calls it Off
The less likely option, but let me play dolls for a minute. In some world, Larson's nerves get the better of him; maybe because he loves his daughter too much, or having absolutely no legitimate human children left to carry his family name is unthinkable, or the idea of being alone forever scares the shit out of him, or a thousand other reasons. But whatever the reason may be, Larson can't go through with it. He calls the entire thing off, leaving Addison deeply confused as to why her father has thrown what seems to be an event of such importance completely out the window for no reason. Larson doesn't tell her, and as she grows older the day fades into memory as an odd, but ultimately insignificant occurrence. Well, that's that. But of course, his ambition remains, and he has to make up for not going through with his oath to the gods somehow. So instead he offers his daughter up to them in another way: he promises he will raise her to become another devoted acolyte in their service, and that she will take his place when he dies. And without the sacrifice for immortality, Larson will die much sooner than he bargained for.
There are two more ways that this could branch off. First, Addison becomes a cultist herself once she is old enough. She took to the knowledge that her father pressed onto her like a duck to water, even when she was young. As she grows, she takes after her father in every way he could have dreamed. She is a faithful and devoted follower, and with Larson's help, she rises in importance in the Order despite it being only for men. She is charming and persuasive, yet ruthless and stubborn in what she has been taught. She has all the ambition that Larson does, and wields it in the same way that he encourages her to. She and Larson have a much better relationship, at least on the surface; Larson is always pleased with her, always happy to offer advice when she asks it (or when she doesn't). The cult has become a really fucked father-daughter bonding activity; between the rituals and sacrifices (by the time she's older, she's well used to it) and all the work they do together, they make a formidable pair. The other members always mutter with envy and a touch of fear about Larson and his daughter, how they are hardly ever seen apart. They share the same features, the same taste for power, the same glint of something unknowable in their dark eyes. Addison sharing her father's goal of serving the gods and achieving some higher purpose facilitates a not-so-healthy bond, but it's a bond nonetheless (and Addison knows nothing different). It's never clear whether this is what Addison truly desires, or if this is the only point of contact that she has with her increasingly invested father. Addison couldn't even say for sure herself.
When Larson dies, it devastates her. After all, he was her guiding hand; how can she go on by herself? The rest of her family is well and truly dead. Yet she still assumes his place in the Order, and ascends in rank year by year until she reaches the point where she becomes the first female grand vizier. She shakes up the Order beyond belief; striking a balance between the merrymaking that its members have become so fond of and the lost original goal that her father always bemoaned. She puts the gears in motion to make the Order co-ed, although it doesn't quite stick while she's in power. Addison does everything her father fell short of and follows his footsteps in a way she hopes would please him. She should be very proud. Her father would be proud of her. Yet for some reason, when she thinks of him, that never seems to be what his voice tells her.
Alternatively, Addison has none of it. She doesn't want to be involved in her father's pet project; maybe she even takes a similar stance of indifference as Arthur towards the gods, no matter how many times Larson drags her to rituals to see them for herself. She simply lacks that penchant for devotion that Larson has. All her stubbornness comes out; Addison purposely fudges rituals, refuses to cooperate when asked, and gives Larson enough back talk to drive him half to madness. Her ambition lies in everything else, it seems; history and science and things of this mortal plane, rather than the arcane studies he tries time and time again to push on her. But Larson's stubborn, too: he never stops trying to rein her in, control what she does and where she goes. He refuses to acknowledge that he could possibly be wrong; if Addison would just cooperate she would understand why his devotion runs so deep. It fulfilled him, and it would do the same for her if she would just listen. Never does he consider she may want to take a different path, after all, why would she? She is his. All deals with the outer gods aside, this is what she should want. Rather than the cold stalemate in option A, it's an ongoing battle between the two of them. They never seem to reach a moment of peace or agreement; just an exhausting back and forth that never seems to end. Addison wants to leave, but maybe she stays out of some family obligation or fear or simply because there is no way out, not one that Larson will let her see. As time passes, every single day she wishes that things would simply go back to the way they were before her father got mixed up in all this and started trying to get her into it as well.
It's only when he dies (of uncertain and brutal causes, as far as Addison can tell. The gods weren't pleased that his promise went unfulfilled for so long), that she can relax. The mines stay open, and Addison lives out the rest of her days in the estate. But unlike her father, she interacts with the community and refuses to stay a recluse. She pushes all the cult stuff behind her and takes responsibility for the town as hers; with the massive fortune left by her father, she provides funds for more businesses and public services outside of the mines (and the college still recieves the observatory on Mt. Hoosac, which she proudly pays for. She always did like astronomy.). The town becomes a thriving, growing community in a way that it never was under Larson's care, and eventually, when the coal runs dry the town remains full of life. Maybe Addison gets married and has children of her own to fill the empty rooms of the estate, or maybe she doesn't. Either way, the estate is never short of friends nor laughter as she causes the town to flourish. Miss Addison Larson is deeply cherished by the people that follow until the day she passes herself, a beloved matriarch, benefactor, and a truly remarkable woman. Larson was right in one respect, anyways: it was her that would make the town. Just not in the way he expected.