I raised this myself several years ago.
Elaine went to Summer for protection, if she stayed in the Never Never she and any child would require mortal food and drink as well as protection, incurring a substantial debt. Firstborn is the gold standard (just ask lea), so she could have traded any child for that and the child’s continued protection.
Aurora may have wanted the child as a future Summer Knight, to balance out Harry, who clearly outpowers the current Summer Knight, and that much is clearer since BG. It always took me that Aurora took her daughters death at Harry’s hand so well. Perhaps because she has Harry’s son and part of that deal was not to harm him?
I wonder if when we see a new Summer Knight Harry takes an instant dislike to him, because he is just like Harry, who only finds out the truth after each tries to kill each other, and he pulls something similar to what Harry did on Eb about Thomas. “Harry, I am your son”.
Whatever the truth we have not heard the full story regarding Elaine.