Unfortunately having been through the last eight years where Obama was required to show nothing officially to prove his status, and courts not being willing to touch it, I’m not nearly so optimistic as you.
One of the glaring ironies of this situation is that the known situation of Cruz - one citizen parent, born abroad - is identical to what Obama’s situation would have been if someone had come up with conclusive evidence that he had been born in Kenya.
Can’t help but wonder, if Cruz folks really did an honesty check, if any of them would say that they would have defended Obama’s NBC qualifications under the same circumstances.