### Apache Airflow version

main (development)

### What happened

https://github.com/apache/airflow/pull/27755 added the triggerer to the rest api health endpoint, but not the main one served on `/health`.

### What you think should happen instead

As documented [here](https://airflow.apache.org/docs/apache-airflow/2.6.1/administration-and-deployment/logging-monitoring/check-health.html#webserver-health-check-endpoint), the `/health` endpoint should include triggerer info like shown on `/api/v1/health`.

### How to reproduce

Compare `/api/v1/health` and `/health` responses.

### Operating System

mac os

### Versions of Apache Airflow Providers

_No response_

### Deployment

Virtualenv installation

### Deployment details

_No response_

### Anything else

_No response_

### Are you willing to submit PR?

- [ ] Yes I am willing to submit a PR!

### Code of Conduct

- [X] I agree to follow this project's [Code of Conduct](https://github.com/apache/airflow/blob/main/CODE_OF_CONDUCT.md)

)
    @expose("/clear", methods=["POST"])
    @auth.has_access(
        [
            (permissions.ACTION_CAN_EDIT, permissions.RESOURCE_DAG),
            (permissions.ACTION_CAN_DELETE, permissions.RESOURCE_TASK_INSTANCE),
        ]
    )
    @action_logging
    @provide_session
    def clear(self, *, session: Session = NEW_SESSION):
        """Clears DAG tasks."""
        dag_id = request.form.get("dag_id")
        task_id = request.form.get("task_id")
        origin = get_safe_url(request.form.get("origin"))
        dag = get_airflow_app().dag_bag.get_dag(dag_id)
        group_id = request.form.get("group_id")
        if "map_index" not in request.form:
            map_indexes: list[int] | None = None
        else:
            map_indexes = request.form.getlist("map_index", type=int)
        execution_date_str = request.form.get("execution_date")
        execution_date = _safe_parse_datetime(execution_date_str)
        confirmed = request.form.get("confirmed") == "true"
        upstream = request.form.get("upstream") == "true"
        downstream = request.form.get("downstream") == "true"
        future = request.form.get("future") == "true"
        past = request.form.get("past") == "true"
        recursive = request.form.get("recursive") == "true"
        only_failed = request.form.get("only_failed") == "true"