### 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
/*
 * Copyright Elasticsearch B.V. and/or licensed to Elasticsearch B.V. under one
 * or more contributor license agreements. Licensed under the Elastic License
 * 2.0; you may not use this file except in compliance with the Elastic License
 * 2.0.
 */
package org.elasticsearch.xpack.shutdown;
import org.elasticsearch.client.Request;
import org.elasticsearch.client.Response;
import org.elasticsearch.client.ResponseException;
import org.elasticsearch.common.Strings;
import org.elasticsearch.common.settings.SecureString;
import org.elasticsearch.common.settings.Settings;
import org.elasticsearch.common.util.concurrent.ThreadContext;
import org.elasticsearch.core.Nullable;
import org.elasticsearch.test.readiness.ReadinessClientProbe;
import org.elasticsearch.test.rest.ESRestTestCase;
import org.elasticsearch.xcontent.ObjectPath;
import org.elasticsearch.xcontent.XContentBuilder;
import org.elasticsearch.xcontent.json.JsonXContent;
import java.io.IOException;
import java.util.List;
import java.util.Map;
import java.util.Optional;
import java.util.concurrent.atomic.AtomicReference;
import static org.hamcrest.Matchers.allOf;
import static org.hamcrest.Matchers.containsString;