The Gauss-Faraday law can be viewed the integrability conditions for the existence of the electro-magnetic $4$-potential $A_{\mu}$ via Poincare Lemma.

Or conversely, assuming a globally defined electro-magnetic $4$-potential $A_{\mu}$, the Gauss-Faraday law is trivially satisfied. See also this related Phys.SE post.

The Gauss-Faraday law can be viewed as the integrability condition for the existence of the electro-magnetic $4$-potential $A_{\mu}$ via Poincare Lemma.

Or conversely, assuming a globally defined electro-magnetic $4$-potential $A_{\mu}$, the Gauss-Faraday law is trivially satisfied. See also this related Phys.SE post.

So yes, Gauss-Faraday law plays a role already before quantization.