### 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)

"""Home view."""
        from airflow.models.dag import DagOwnerAttributes
        hide_paused_dags_by_default = conf.getboolean("webserver", "hide_paused_dags_by_default")
        default_dag_run = conf.getint("webserver", "default_dag_run_display_number")
        num_runs = request.args.get("num_runs", default=default_dag_run, type=int)
        current_page = request.args.get("page", default=0, type=int)
        arg_search_query = request.args.get("search")
        arg_tags_filter = request.args.getlist("tags")
        arg_status_filter = request.args.get("status")
        arg_sorting_key = request.args.get("sorting_key", "dag_id")
        arg_sorting_direction = request.args.get("sorting_direction", default="asc")