Birth records are sealed naturally to curtail identity theft and protect privacy. I can't get a copy of your birth certificate without your consent. You can look up "how to get a copy of your birth certificate" at any birth records registrar, they have similar requirements for getting a certified copy of your birth certificate.
As for why Cruz released his birth certificate, I assume it was to provide at least part of the evidence that he was a citizen at birth. His BC shows his mother born in the US. I agree with those who say this is not sufficient evidence to establish citizenship, but it is part of that package.
Why not the CRBA?