ingress-nginx doesn't do readiness/liveness checks itself; the kubelet does[1].
If any of the containers in a Pod aren't ready, the endpoints controller[2] removes the Endpoint object corresponding to that Pod. ingress-nginx watches these Endpoints objects[3] to determine which Pods it should send traffic to.
Edit: As to your use case, I think you should remove the readiness probe from that one container you don't care about.