### 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
.filter(decider -> "node_shutdown".equals(decider.get("decider")))
                .allMatch(decider -> "NO".equals(decider.get("decision"))),
            is(true)
        );
    }
    /**
     * Checks that shards properly move off of a node that's marked for removal, including:
     * 1) A reroute needs to be triggered automatically when the node is registered for shutdown, otherwise shards won't start moving
     *    immediately.
     * 2) Ensures the status properly comes to rest at COMPLETE after the shards have moved.
     */
    @SuppressWarnings("unchecked")
    @AwaitsFix(bugUrl = "https://github.com/elastic/elasticsearch/issues/77488")
    public void testShardsMoveOffRemovingNode() throws Exception {
        String nodeIdToShutdown = getRandomNodeId();
        final String indexName = "test-idx";
        Request createIndexRequest = new Request("PUT", indexName);
        createIndexRequest.setJsonEntity("{\"settings\":  {\"number_of_shards\": 4, \"number_of_replicas\": 0}}");
        assertOK(client().performRequest(createIndexRequest));
        ensureGreen(indexName);
        
        Request checkShardsRequest = new Request("GET", "_cat/shards/" + indexName);
        checkShardsRequest.addParameter("format", "json");
        checkShardsRequest.addParameter("h", "index,shard,prirep,id,state");
        
        {
            List<Object> shardsResponse = entityAsList(client().performRequest(checkShardsRequest));
            final long shardsOnNodeToShutDown = shardsResponse.stream()
                .map(shard -> (Map<String, Object>) shard)
                .filter(shard -> nodeIdToShutdown.equals(shard.get("id")))