I won't lie - I have had the occasional wobble myself about all of this.
But I just don't think Egon would have left that house to someone he didn't know and trust to some degree. I mean, he's not stupid. He knows what's at that house, and yeah, I know it's all hidden away but still. He knows the danger in that area. He knows what's probably on the horizon so he'd leave it to someone he trusts to deal with the potential fallout at some point. Or, at least someone who already knows who to call when the fallout does happen.
This is what in my opinion blows the theory out of the water that a) Callie doesn't know her father and he doesn't know her, and that b) Callie's mother was either someone random he knocked up while in college or someone he donated sperm to (which, IMO, is the most stupidest theory of them all, because sperm donation should by all accounts be anonymous so how he even knew about Callie let alone chose her among the probably hundreds/thousands of other kids he's fathered through donation just does. not. make. any. sense what-so-freaking-ever!!!!)
'They have left everything they know and moved to a small town in Oklahoma after inheriting property from the father she didn’t know.
' - Vanity Fair article
Yes, it does imply that Egon was a deadbeat dad. But it also could imply that Egon passed away when she was small and she never got a chance to get to know him.
First, let's go with the deadbeat dad theory, which I suppose is how you would take that article's sentence in the most literal sense.
Let's say he hooked up with some random student while he was in college or whatever, and knocked her up. He didn't know this kid existed until all of a sudden this random science chick who dug his epidiymus makes all the effort to find him waaaay out in the middle of nowheresville Oklahoma to tell him has a daughter somewhere, and he decides not to bother getting to know his daughter at all, leaves her and her mother in Chicago while he continues to do his thing, and then when he's writing his will he remembers to leave his house to her, filled to the brim with all his ghostbusters stuff, which conveniently is located very close to a mine which is suspiciously named for a guy who he knows opened a door to another dimension bringing in a deity that almost destroyed the City of New York. Gee, thanks, Dad.
According to this theory, if Egon was an a-hole who wasn't going to be a part of his kid's life voluntarily, why would he then decide to leave all of his belongings to her, including pretty dangerous equipment that he knew shouldn't be in the hands of people that he didn't trust or know well, on a site that has the potential to be ground zero for some pretty dangerous shit going down at some point? How would he know that this child of his won't sell all the crap they find and move elsewhere with the profits? How does he know that this child of his will be able to deal with the potential fallout from whatever might happen in Summersville?
Egon has taken the time and effort to locate this mine, buy property close to it so that he could keep an eye on it, hide all of his equipment on the property in case of emergency, and now he is just handing it over to someone he doesn't know when Ray Stantz is a friend, ex-colleague, and would know exactly what to do in the situation Egon has spent so much time monitoring. If Egon was alone, had no close family that he knew of, although knew he had an illegitimate child somewhere but didn't bother to get to know her, why would he chose to leave all of that to her rather than to Ray, a trusted friend and ex-colleague?
Now, let's go with the theory that Egon sadly passed away when Callie was too young to know her father.
There's two versions of this theory. Version 1) Callie's mother was a random ex-student of his from back in University. A science chick who was interested in his epididymus. They hook up again sometime after the events of GB2 while Callie is very young, long enough for Egon to get to know his daughter a little. But he passes away while she's still a young child.
This is where Callie's mother's role becomes important. Again, if he didn't trust Callie to take on the responsibility of that house and everything inside it, then he'd leave it to Ray, wouldn't he? He would have to also trust this science chick with the contents of the house, enough to assist Callie with it after he's passed away. He expects Callie's mother to be there with Callie after he's passed away. She's probably younger than him, after all. I think it's the same with most families. You'd expect the other half to be around to help deal with things after you pass on. But if this science student knows nothing about the business, knows nothing about all the equipment, why again would he leave the house to them, placing them both potentially in a hell of a lot of danger?
Unless... Version 2) Callie's mother was Janine Melnitz. Former ghostbusters secretary. Someone who knows the business like the back of her hand. Someone who knows what the equipment does, and how it is used. Someone who knows when the shit does go down, who they would need to call... If Janine was bringing up their child, he'd expect her to have told Callie about the business, maybe let them know about the house and its secrets, and most importantly, let Callie know who to call when the proverbial hits the fan.
Why Janine is absent or didn't do any of this is unclear. Maybe she was taken ill. Maybe she's incapacitated somehow. Maybe she was targeted by someone. Maybe people think she's crazy and sent her to a retirement home because she's talking about strange mines in the middle of Oklahoma somewhere...
The fact remains that IMO Egon would know Callie existed, know her well enough to trust her (or someone she knows well) with dangerous equipment and the potential fallout from the mine's activity, otherwise he'd have passed that house onto someone else.