### 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)

"""Returns grid data."""
        dag_id = request.args.get("dag_id")
        dag = get_airflow_app().dag_bag.get_dag(dag_id)
        if not dag:
            return {"error": f"can't find dag {dag_id}"}, 404
        root = request.args.get("root")
        if root:
            filter_upstream = request.args.get("filter_upstream") == "true"
            filter_downstream = request.args.get("filter_downstream") == "true"
            dag = dag.partial_subset(
                task_ids_or_regex=root, include_upstream=filter_upstream, include_downstream=filter_downstream
            )
        num_runs = request.args.get("num_runs", type=int)
        if num_runs is None:
            num_runs = conf.getint("webserver", "default_dag_run_display_number")
        try:
            base_date = timezone.parse(request.args["base_date"])
        except (KeyError, ValueError):
            base_date = dag.get_latest_execution_date() or timezone.utcnow()
        with create_session() as session:
            query = session.query(DagRun).filter(
                DagRun.dag_id == dag.dag_id, DagRun.execution_date <= base_date
            )
            run_type = request.args.get("run_type")
            if run_type:
                query = query.filter(DagRun.run_type == run_type)
            run_state = request.args.get("run_state")
            if run_state: