### Description

The new [readiness port](https://www.elastic.co/guide/en/elasticsearch/reference/8.4/advanced-configuration.html#readiness-tcp-port) makes readiness checks on Kubernetes much easier. However the current implementation only allows connections through localhost. TCP probes on Kubernetes are run by the kubelet which is a process running on the Kubernetes node (and not from within the container or the Pod). The [Kubernetes documentation](https://kubernetes.io/docs/tasks/configure-pod-container/configure-liveness-readiness-startup-probes/#tcp-probes) mentions it, but it is kind of subtle: 

> For a TCP probe, the kubelet makes the probe connection at the node, not in the pod, which means that you can not use a service name in the host parameter since the kubelet is unable to resolve it.


That means a standard Kubernetes readiness probe cannot be used for Elasticsearch:
```
          readinessProbe:
            tcpSocket:
              port: 8080
```

Instead, only an `exec` probe that runs a shell command inside the container can be used that relies in this case on the presence of the `nc` command in the container image:

```
    readinessProbe:
      exec:
        command: ["nc", "-z", "-v", "-w5", "127.0.0.1", "8080"] 
```



x-pack/plugin/shutdown/qa/multi-node/src/javaRestTest/java/org/elasticsearch/xpack/shutdown/NodeShutdownIT.java
putNodeShutdown(nodeIdToShutdown, "remove");
        {
            
            Request getStatusRequest = new Request("GET", "_nodes/" + nodeIdToShutdown + "/shutdown");
            Response statusResponse = client().performRequest(getStatusRequest);
            Map<String, Object> status = entityAsMap(statusResponse);
            assertThat(ObjectPath.eval("nodes.0.shard_migration.status", status), equalTo("STALLED"));
            assertThat(ObjectPath.eval("nodes.0.shard_migration.shard_migrations_remaining", status), equalTo(numberOfShards));
            assertThat(
                ObjectPath.eval("nodes.0.shard_migration.explanation", status),
                allOf(
                    containsString(indexName),
                    containsString("cannot move, see [node_allocation_decision] for details or use the cluster allocation explain API")
                )
            );
            assertThat(ObjectPath.eval("nodes.0.shard_migration.node_allocation_decision", status), notNullValue());
        }
        
        Request updateSettingsRequest = new Request("PUT", indexName + "/_settings");
        updateSettingsRequest.setJsonEntity("{\"index.routing.allocation.require._id\": null}");
        assertOK(client().performRequest(updateSettingsRequest));
        assertBusy(() -> {
            Request getStatusRequest = new Request("GET", "_nodes/" + nodeIdToShutdown + "/shutdown");
            Response statusResponse = client().performRequest(getStatusRequest);
            Map<String, Object> status = entityAsMap(statusResponse);
            assertThat(ObjectPath.eval("nodes.0.shard_migration.status", status), equalTo("COMPLETE"));
            assertThat(ObjectPath.eval("nodes.0.shard_migration.shard_migrations_remaining", status), equalTo(0));
            assertThat(ObjectPath.eval("nodes.0.shard_migration.explanation", status), nullValue());
        });
    }