### 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)

"This indicates that this deployment may be accessible to the public internet. "
                        "This warning can be disabled by setting webserver.warn_deployment_exposure=False in "
                        "airflow.cfg. Read more about web deployment security <a href="
                        f'"{get_docs_url("security/webserver.html")}">'
                        "here</a>"
                    ),
                    "warning",
                )
        return self.render_template(
            "airflow/dags.html",
            dags=dags,
            dashboard_alerts=dashboard_alerts,
            migration_moved_data_alerts=sorted(set(_iter_parsed_moved_data_table_names())),
            current_page=current_page,
            search_query=arg_search_query if arg_search_query else "",
            page_title=Markup(page_title) if page_title_has_markup else page_title,
            page_size=dags_per_page,
            num_of_pages=num_of_pages,
            num_dag_from=min(start + 1, num_of_all_dags),
            num_dag_to=min(end, num_of_all_dags),
            num_of_all_dags=num_of_all_dags,
            paging=wwwutils.generate_pages(
                current_page,
                num_of_pages,
                search=escape(arg_search_query) if arg_search_query else None,
                status=arg_status_filter if arg_status_filter else None,
                tags=arg_tags_filter if arg_tags_filter else None,
                sorting_key=arg_sorting_key if arg_sorting_key else None,
                sorting_direction=arg_sorting_direction if arg_sorting_direction else None,
            ),