[thackney]

To obtain his US citizenship he had to file paperwork, pay a fee and go through a process.

What paperwork beyond a record of his birth? What process? Did it require someone to approve, or just a record of birth?

[Cboldt]

The process results in either a Consular Record for Birth Abroad (CRBA, FS-240), or a Certification of Citizenship. Both processes are an examination of the evidence, and if the evidence supports finding the statutory conditions are satisfied, the credential is issued. It's not "approval," just review.

Even if the credential isn't sought (but Cruz has a CRBA, I take as a matter of informed faith), the same conclusion of "citizen at birth by operation of a statute" would be found by a competent court, on evidence that the statutory conditions were met.