It's not fully answered. Motherhood gave a purpose to Kate, but Sun still spent most of her efforts avenging Jin, not happily raising Ji Yeon. And then, she learned Jin was alive, and she could be happy no more. She left her daughter behind to go find Jin.

I think Jacob thought if one of these two accepted to become the guardian, the other would stay as well.

But indeed it is more likely Jin Jacob still had on the list. But like Kate the job was probably still opened to Sun if she wanted it.