### Elasticsearch Version

7.16.2

### Installed Plugins

_No response_

### Java Version

_bundled_

### OS Version

Cloud

### Problem Description

When using the `final_pipeline` option and the `date match index` in a pipeline to process data, the final_pipeline appears to run twice.  This can be shown by using the `append` processor which adds two of the same values or a `rename` processor which shows a failure, but it will show the name of a field changed and shows a failure due to the `old` named field not being present.

Also referenced in a [discuss topic](https://discuss.elastic.co/t/with-final-pipeline-and-a-pipeline-specified-the-final-pipeline-appears-to-be-running-twice-with-index-date-math-processor/296593) to confirm that the example is valid.

Additional note: This seems to run fine on a 7.6.2 environment, so something between that and 7.16.2 may have caused this issue.  #69727 or #75047 could be related.

### Steps to Reproduce

Append method of showing the `final_pipeline` running twice:

```
PUT _ingest/pipeline/test-final-pipeline
{
  "processors": [
    {
      "append": {
        "field": "pipeline",
        "value": "completed"
      }
    }
  ]
}

PUT _template/test-template
{
  "index_patterns": [
    "test*"
  ],
  "settings": {
    "final_pipeline": "test-final-pipeline",
    "number_of_shards": 2
  }
}

POST /test-1/_doc
{
  "time-field": "2021-02-10T12:02:01.789Z",
  "CONTENT": "blah1"
}

PUT /_ingest/pipeline/routing-test-pipeline
{
  "description": "Time series DAY pipeline",
  "processors": [
     {
         "date_index_name": {
             "field": "time-field",
             "index_name_prefix": "test-",
             "date_rounding": "d",
             "index_name_format": "yyyy-MM-dd"
         }
     }
 ]
}

POST /test-1/_doc?pipeline=routing-test-pipeline
{
  "time-field": "2021-02-10T12:02:01.789Z",
  "CONTENT": "blah2"
}
```

Rename processor used to show a failure.  
```
PUT _ingest/pipeline/test-final-pipeline
{
  "processors": [
    {
      "rename": {
        "field": "CONTENT",
        "target_field": "MALCONTENT"
      }
    }
  ]
}

PUT _template/test-template
{
  "index_patterns": [
    "test*"
  ],
  "settings": {
    "final_pipeline": "test-final-pipeline",
    "number_of_shards": 2
  }
}

POST /test-1/_doc
{
  "time-field": "2021-02-10T12:02:01.789Z",
  "CONTENT": "blah1"
}

PUT /_ingest/pipeline/routing-test-pipeline
{
  "description": "Time series DAY pipeline",
  "processors": [
     {
         "date_index_name": {
             "field": "time-field",
             "index_name_prefix": "test-",
             "date_rounding": "d",
             "index_name_format": "yyyy-MM-dd"
         }
     }
 ]
}

POST /test-1/_doc?pipeline=routing-test-pipeline
{
  "time-field": "2021-02-10T12:02:01.789Z",
  "CONTENT": "blah2"
}
```

### Logs (if relevant)

_No response_
server/src/main/java/org/elasticsearch/ingest/IngestService.java
indexRequest.id()
                            ),
                            ex
                        )
                    );
                    return; 
                }
                Iterator<String> newPipelineIds = pipelineIds;
                boolean newHasFinalPipeline = hasFinalPipeline;
                final String newIndex = indexRequest.indices()[0];
                if (Objects.equals(originalIndex, newIndex) == false) {
                    if (hasFinalPipeline && pipelineIds.hasNext() == false) {
                        listener.onFailure(
                            new IllegalStateException(
                                format(
                                    "final pipeline [%s] can't change the target index (from [%s] to [%s]) for document [%s]",
                                    pipelineId,
                                    originalIndex,
                                    newIndex,
                                    indexRequest.id()
                                )
                            )
                        );
                        return; 
                    } else {
                        indexRequest.isPipelineResolved(false);
                        resolvePipelinesAndUpdateIndexRequest(null, indexRequest, state.metadata());
                        if (IngestService.NOOP_PIPELINE_NAME.equals(indexRequest.getFinalPipeline()) == false) {
                            newPipelineIds = Collections.singleton(indexRequest.getFinalPipeline()).iterator();
                            newHasFinalPipeline = true;