I wonder if “no choice” might be better than “no part” only because it’s currently got a tiny bit of implication that they adopted her. This way we know her mother gave birth to her. Plus we open the door for her to still be genetically related to her mother, as assumably Anakin is related to Shmi by genetics as well. It would just be Rey’s father who would be the odd guy out. Another added benefit is that it implies the same thing as them never wanting her without directly saying it.

EDIT: Eh, I’ll probably just stick with “no part” only because any confusion would be cleared up later and it’s best to start with a significant level of intrigue.