Of course, adding the axiom ZFC is consistent to the theory ZFC does _not_ produce a convincing proof of the consistency of ZFC!
The theory Ordinals has a convincing proof of consistency because the theory has a provably unique up to isomorphism model. Consequently, ZFC must also be consistent because it is a special case of the theory Ordinals.
The theory Ordinals has a convincing proof of consistency because the theory has a provably unique up to isomorphism model. Consequently, ZFC must also be consistent because it is a special case of the theory Ordinals.