### 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)

current_dags = current_dags.order_by(null_case, dag_run_subquery.c.max_execution_date)
            else:
                sort_column = DagModel.__table__.c.get(arg_sorting_key)
                if sort_column is not None:
                    null_case = case((sort_column.is_(None), 1), else_=0)
                    if arg_sorting_direction == "desc":
                        current_dags = current_dags.order_by(null_case, sort_column.desc())
                    else:
                        current_dags = current_dags.order_by(null_case, sort_column)
            dags = current_dags.options(joinedload(DagModel.tags)).offset(start).limit(dags_per_page).all()
            user_permissions = g.user.perms
            can_create_dag_run = (
                permissions.ACTION_CAN_CREATE,
                permissions.RESOURCE_DAG_RUN,
            ) in user_permissions
            dataset_triggered_dag_ids = {dag.dag_id for dag in dags if dag.schedule_interval == "Dataset"}
            if dataset_triggered_dag_ids:
                dataset_triggered_next_run_info = get_dataset_triggered_next_run_info(
                    dataset_triggered_dag_ids, session=session
                )
            else:
                dataset_triggered_next_run_info = {}
            for dag in dags:
                dag.can_edit = get_airflow_app().appbuilder.sm.can_edit_dag(dag.dag_id, g.user)
                dag.can_trigger = dag.can_edit and can_create_dag_run
                dag.can_delete = get_airflow_app().appbuilder.sm.can_delete_dag(dag.dag_id, g.user)
            dagtags = session.query(func.distinct(DagTag.name)).order_by(DagTag.name).all()
            tags = [
                {"name": name, "selected": bool(arg_tags_filter and name in arg_tags_filter)}
                for name, in dagtags