### Elasticsearch Version

7.17, 8.x

### Installed Plugins

_No response_

### Java Version

_bundled_

### OS Version

Darwin

### Problem Description

The migrate to data tiers routing API changes the allocation configuration for 
the indices that need migrating to data tiers routing. 
After such changes we should do a cluster re-route so the allocation changes
take effect immediately. 

### Steps to Reproduce

1. On a one node `data_hot,data_content,master` node create the following index
```
PUT datatiers_problem_index
{
    "settings": {
        "index.routing.allocation.include._tier_preference": "data_hot",
        "index.routing.allocation.require.data": "hot"
    }
}
```
2. Migrate to data tiers and observe the shard is not allocated afterwards
`POST _ilm/migrate_to_data_tiers`

### Logs (if relevant)

_No response_
x-pack/plugin/ilm/qa/multi-node/src/javaRestTest/java/org/elasticsearch/xpack/MigrateToDataTiersIT.java
private String index;
    private String policy;
    private String alias;
    @Before
    public void refreshIndexAndStartILM() throws IOException {
        index = "index-" + randomAlphaOfLength(10).toLowerCase(Locale.ROOT);
        policy = "policy-" + randomAlphaOfLength(5);
        alias = "alias-" + randomAlphaOfLength(5);
        assertOK(client().performRequest(new Request("POST", "_ilm/start")));
    }
    @AfterClass
    public static void restartIlm() throws IOException {
        
        assertOK(client().performRequest(new Request("POST", "_ilm/start")));
    }
    public void testAPIFailsIfILMIsNotStopped() throws IOException {
        Request migrateRequest = new Request("POST", "_ilm/migrate_to_data_tiers");
        ResponseException e = expectThrows(ResponseException.class, () -> client().performRequest(migrateRequest));
        assertThat(e.getResponse().getStatusLine().getStatusCode(), is(RestStatus.INTERNAL_SERVER_ERROR.getStatus()));
        assertThat(e.getMessage(), containsString("stop ILM before migrating to data tiers, current state is [RUNNING]"));
    }
    @SuppressWarnings("unchecked")