According to US Code, a CRBA is the same as a certificate of naturalization. It is not listed with the passports, it is listed with that form of citizenship that is pursuant to regulation.
Natural born citizens are not pursuant to regulation as no law governs their nationality at birth, they come by it by the fact that at birth, they were not citizens of any foreign country. And the mere fact that a CRBA can be denied, well, that alone tells us that being born abroad to one citizen parent does not automatically make one a US citizen at birth. The CRBA is an application for citizenship, it does not replace ones birth certificate, it merely makes it easier for ones US citizenship to be perfected at a later date by being able to by-pass the full process of naturalization.
A CRBA is NOT an application for citizenship. It’s an application for PROOF of citizenship and yes, it can be denied if your child is not, in fact, born a US citizen. There is no law that you have to have a CRBA if you are born abroad. It helps people that must a apply for a passport for their child to exit the country of their birth. Cruz did not need a passport to exit Canada when he was 4 years old.