### 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)

session.query(DagRun.dag_id, func.max(DagRun.start_date).label("start_date"))
                .filter(DagRun.state == State.FAILED)
                .group_by(DagRun.dag_id)
                .subquery()
            )
            subq_join = (
                session.query(subq_all.c.dag_id, subq_all.c.start_date)
                .join(
                    subq_failed,
                    and_(
                        subq_all.c.dag_id == subq_failed.c.dag_id,
                        subq_all.c.start_date == subq_failed.c.start_date,
                    ),
                )
                .subquery()
            )
            failed_dags = dags_query.join(subq_join, DagModel.dag_id == subq_join.c.dag_id)
            is_paused_count = dict(
                all_dags.with_entities(DagModel.is_paused, func.count(DagModel.dag_id))
                .group_by(DagModel.is_paused)
                .all()
            )
            status_count_active = is_paused_count.get(False, 0)
            status_count_paused = is_paused_count.get(True, 0)
            status_count_running = running_dags.count()
            status_count_failed = failed_dags.count()
            all_dags_count = status_count_active + status_count_paused
            if arg_status_filter == "active":
                current_dags = active_dags
                num_of_all_dags = status_count_active