### 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(shard -> "STARTED".equals(shard.get("state")))
                .count();
            assertThat(shardsOnNodeToShutDown, is(1L));
        }
        
        putNodeShutdown(nodeIdToShutdown, "REMOVE");
        
        AtomicReference<List<Object>> debug = new AtomicReference<>();
        assertBusy(() -> {
            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")))
                .filter(shard -> "STARTED".equals(shard.get("state")) || "RELOCATING".equals(shard.get("state")))
                .count();
            assertThat(shardsOnNodeToShutDown, is(0L));
            debug.set(shardsResponse);
        });
        
        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());
    }
    public void testShardsCanBeAllocatedAfterShutdownDeleted() throws Exception {
        String nodeIdToShutdown = getRandomNodeId();
        putNodeShutdown(nodeIdToShutdown, "REMOVE");