### 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)

escaped_arg_search_query = arg_search_query.replace("_", r"\_")
                dags_query = dags_query.filter(
                    DagModel.dag_id.ilike("%" + escaped_arg_search_query + "%", escape="\\")
                    | DagModel.owners.ilike("%" + escaped_arg_search_query + "%", escape="\\")
                )
            if arg_tags_filter:
                dags_query = dags_query.filter(DagModel.tags.any(DagTag.name.in_(arg_tags_filter)))
            dags_query = dags_query.filter(DagModel.dag_id.in_(filter_dag_ids))
            filtered_dag_count = dags_query.count()
            if filtered_dag_count == 0 and len(arg_tags_filter):
                flash(
                    "No matching DAG tags found.",
                    "warning",
                )
                flask_session[FILTER_TAGS_COOKIE] = None
                return redirect(url_for("Airflow.index"))
            all_dags = dags_query
            active_dags = dags_query.filter(~DagModel.is_paused)
            paused_dags = dags_query.filter(DagModel.is_paused)
            
            running_dags = dags_query.join(DagRun, DagModel.dag_id == DagRun.dag_id).filter(
                DagRun.state == State.RUNNING
            )
            
            subq_all = (
                session.query(DagRun.dag_id, func.max(DagRun.start_date).label("start_date"))
                .group_by(DagRun.dag_id)
                .subquery()
            )
            subq_failed = (