I don't think Alice was simple minded. I do think she was probably a very kind hearted person who was manipulated by a clever con merchant.
I have been manipulated by somebody way less clever than Anna, because they played on my emotions and knew -exactly- what strings to pull. Did I feel like an idiot later? Yes, I sure did.. because I am pretty darned smart. Still, intellectual smart is not emotionally smart, and I can easily see how a childless woman could come to love a child as her own, and his mother as her dearest friend.. and how that love could be used as leverage.
Alice was duped by people she trusted, and loved, IMO. It happens to the best of us.
Anna was a rather romantic and sympathetic figure - Russian nobility, single mother of a half-Asian child, a tragic widow forced into domestic work by circumstance, and rather more worldly than Alice was ever likely to be. I can really see how the con went.. and how Alice was duped. Poor woman.