### 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
import static org.hamcrest.Matchers.empty;
import static org.hamcrest.Matchers.equalTo;
import static org.hamcrest.Matchers.equalToIgnoringCase;
import static org.hamcrest.Matchers.hasSize;
import static org.hamcrest.Matchers.is;
import static org.hamcrest.Matchers.not;
import static org.hamcrest.Matchers.notNullValue;
import static org.hamcrest.Matchers.nullValue;
public class NodeShutdownIT extends ESRestTestCase implements ReadinessClientProbe {
    public void testRestartCRUD() throws Exception {
        checkCRUD(randomFrom("restart", "RESTART"), randomPositiveTimeValue(), null);
    }
    public void testRemoveCRUD() throws Exception {
        checkCRUD(randomFrom("remove", "REMOVE"), null, null);
    }
    public void testReplaceCRUD() throws Exception {
        checkCRUD(randomFrom("replace", "REPLACE"), null, randomAlphaOfLength(10));
    }
    public void checkCRUD(String type, @Nullable String allocationDelay, @Nullable String targetNodeName) throws Exception {
        String nodeIdToShutdown = getRandomNodeId();
        checkCRUD(nodeIdToShutdown, type, allocationDelay, targetNodeName, true);
    }
    @SuppressWarnings("unchecked")
    public void checkCRUD(
        String nodeIdToShutdown,
        String type,
        @Nullable String allocationDelay,
        @Nullable String targetNodeName,
        boolean delete
    ) throws Exception {