You are right that some differential privacy proofs have later been found to be wrong. For example, there is an entire paper about bugs in initial versions of the sparse vector technique [1].
However, I imagine this will evolve the way cryptographic security has evolved: at some point, enough experts have examined algorithm X to be confident about its differential privacy proof; then some experts implement it carefully; and the rest of us use their work because "rolling [our] own" is too tricky.