It suddenly occurs to me that the first woman is the right choice for raising the child, regardless of who the birth mother is.

Indeed; I strongly suspect Solomon had that in mind, but wanted to keep the post as short as possible.

I wonder if Solomon had plans in mind if both women had said the same thing.

Quite possibly. I also wonder if it would depend on what they both said- if both volunteered to retract their claim, then as wedrifid suggests lots were commonly used to show the will of God. If both reacted spitefully, then...