### Elasticsearch Version

8.12

### Installed Plugins

_No response_

### Java Version

_bundled_

### OS Version

N/A

### Problem Description

I was fixing an error that omitted the elastic product header checks for V8 (https://github.com/elastic/elasticsearch/blob/8be04463e4ae5795fc3fad45f2d01314eaf81035/test/yaml-rest-runner/src/main/java/org/elasticsearch/test/rest/yaml/section/DoSection.java#L366-L369).
However, by doing that [WatcherYamlRestIT » test {p0=watcher/update_settings/10_update_watcher_settings/Test update and get watch settings api}](https://gradle-enterprise.elastic.co/s/h4tfsqnf7aajk/tests/task/:x-pack:plugin:watcher:qa:rest:yamlRestTest/details/org.elasticsearch.smoketest.WatcherYamlRestIT/test%20%7Bp0%3Dwatcher%2Fupdate_settings%2F10_update_watcher_settings%2FTest%20update%20and%20get%20watch%20settings%20api%7D) starts to fail.
The "X-elastic-product" header is really missing from the headers. I suspect this bug is due to these lines:
https://github.com/elastic/elasticsearch/blob/f77c16b78bab5e9f3ee24974aeba408e254e8c94/x-pack/plugin/watcher/src/main/java/org/elasticsearch/xpack/watcher/transport/actions/TransportGetWatcherSettingsAction.java#L65-L73

That clear up the context in the response.

### Steps to Reproduce

The bug is present in all versions from 8.8.0 to current; to trigger it is sufficient to call the API and observer the resulting response, and see that the product header is not there.

A way to do that is to remove the `if` here (https://github.com/elastic/elasticsearch/blob/8be04463e4ae5795fc3fad45f2d01314eaf81035/test/yaml-rest-runner/src/main/java/org/elasticsearch/test/rest/yaml/section/DoSection.java#L366-L369) and run linked failing tests, e.g. 
```./gradlew ':x-pack:plugin:watcher:qa:rest:yamlRestTest' --tests "org.elasticsearch.smoketest.WatcherYamlRestIT.test {p0=watcher/update_settings/10_update_watcher_settings/Test update and get watch settings api}" ```

### Logs (if relevant)

_No response_
x-pack/plugin/watcher/src/main/java/org/elasticsearch/xpack/watcher/transport/actions/TransportGetWatcherSettingsAction.java
GetWatcherSettingsAction.Request,
    GetWatcherSettingsAction.Response> {
    @Inject
    public TransportGetWatcherSettingsAction(
        TransportService transportService,
        ClusterService clusterService,