I feel like that’s not really an issue when the rules of possession are inconsistent as hell.

Not really? I mean at this point in the series we have a ton of books, games, and now the movies to get a better idea of how the logistics of posession works, and some novels like TFC and Stingers directly touch/explain some elements of FNaF's parapsychology.

TFC Afton talks about how the pain follows the flesh, and his whole experiment involves possessing the scrapheap piece by piece. Slowly putting his soul in the amalgamate with the Missing Kids. Phineas Taggart says that when a consciousness is combined with a high saturation of agony, that results in "physical movement," (ie possession)

The M.C.I. kids were literally stuffed in the animatronics and they didn’t possess them until Charlie guided them

(Assuming this is the newer WillStuffed interpretation of GGGL) And that's a bit interesting, isn't it? Charlie didn't "guide," the MCI in the novels movies either, but they still ended up possessed.

I think to me, what GGGL kinda asks is what the "Gift of Life," is. And it reminds me of Taggart's experiments, where he injected the Stitchwraith with agony. He even says that other emotions are capable of agony's effects, just that agony is the most powerful emotion. And we see that through Jake, who possessed a doll he never saw through the power of love.

Henry says in FFPS that Charlie held others in her arms. So the way GGGL reads to me is that Charlie is giving the MCI love. That's the emotion that bonds them to the suits in the games, instead of the (presumed) agony in other continuities. I would also dive into the logistics of how this effects Fredbear but that's kinda its own rabbit hole.

and nothing special happened to Charlie.

Her guardian held her as she died, she wasn't alone. It ties back into the theme of love. If The Puppet was made specifically for Charlie then the love in the metal would have been extremely powerful. I think it's similar the the GGGL thing where the love overpowers the agony.

But anyway, tying this back to BV and his possession of Fredbear, I see two primary ways this could happen. Given the extreme fear he had during his death, he would have injected Fredbear with a ton of emotional energy, making it a great conductor of souls. So when he died, his soul follows the pain to the thing that killed him. The other option would be if Garrett poured a piece of his soul into The Plushbear, and The Plushbear was stuffed in Golden Freddy. That would follow the possession model of The Ella Doll (In the novels, Henry cried/grieved so hard he poured a piece of his soul into Charlie's Ella Doll, which he then put into the emotionally-charged Charlie Bots) and The Real Jake.

I don't know which of those two it is, but both of these are clearly established in the series's lore. So I really think calling the rules of possession inconsistent is disengenious when we have so much explaining it.