No, you don't need them all up, but you do need more than half of them to be running.
By default a Kubernetes StatefulSet will start the pods up one at a time and will only start each pod once the previous one is healthy. You can alter this behaviour to avoid the probes by setting podManagementPolicy to Parallel, or else you can account for this in your readiness probe implementation: a pod can be considered ready once it responds to GET /. The Helm chart implements this latter idea.