### 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)

else:
            final_task_instance_query_result = running_task_instance_query_result.subquery("final_ti")
        qry = (
            session.query(
                final_task_instance_query_result.c.dag_id,
                final_task_instance_query_result.c.state,
                final_task_instance_query_result.c.is_dag_running,
                sqla.func.count(),
            )
            .group_by(
                final_task_instance_query_result.c.dag_id,
                final_task_instance_query_result.c.state,
                final_task_instance_query_result.c.is_dag_running,
            )
            .order_by(
                final_task_instance_query_result.c.dag_id,
                final_task_instance_query_result.c.is_dag_running.desc(),
            )
        )
        data = get_task_stats_from_query(qry)
        payload: dict[str, list[dict[str, Any]]] = collections.defaultdict(list)
        for dag_id in filter_dag_ids:
            for state in State.task_states:
                count = data.get(dag_id, {}).get(state, 0)
                payload[dag_id].append({"state": state, "count": count})
        return flask.json.jsonify(payload)
    @expose("/last_dagruns", methods=["POST"])
    @auth.has_access(
        [
            (permissions.ACTION_CAN_READ, permissions.RESOURCE_DAG),