### 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
final String indexName = "test-idx";
        Request createIndexRequest = new Request("PUT", indexName);
        createIndexRequest.setJsonEntity("{\"settings\":  {\"number_of_shards\": 1, \"number_of_replicas\": 3}}");
        assertOK(client().performRequest(createIndexRequest));
        
        assertUnassignedShard(nodeIdToShutdown, indexName);
        
        Request deleteRequest = new Request("DELETE", "_nodes/" + nodeIdToShutdown + "/shutdown");
        assertOK(client().performRequest(deleteRequest));
        assertNoShuttingDownNodes(nodeIdToShutdown);
        
        ensureGreen(indexName);
    }
    @AwaitsFix(bugUrl = "https://github.com/elastic/elasticsearch/issues/77456")
    public void testStalledShardMigrationProperlyDetected() throws Exception {
        String nodeIdToShutdown = getRandomNodeId();
        int numberOfShards = randomIntBetween(1, 5);
        
        final String indexName = "test-idx";
        Request createIndexRequest = new Request("PUT", indexName);
        createIndexRequest.setJsonEntity(formatted("""
            {
              "settings": {
                "number_of_shards": %s,
                "number_of_replicas": 0,
                "index.routing.allocation.require._id": "%s"
              }
            }""", numberOfShards, nodeIdToShutdown));
        assertOK(client().performRequest(createIndexRequest));