### 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)

"""Redirects to external log."""
        dag_id = request.args.get("dag_id")
        task_id = request.args.get("task_id")
        execution_date = request.args.get("execution_date")
        dttm = _safe_parse_datetime(execution_date)
        map_index = request.args.get("map_index", -1, type=int)
        try_number = request.args.get("try_number", 1)
        ti = (
            session.query(models.TaskInstance)
            .filter_by(dag_id=dag_id, task_id=task_id, execution_date=dttm, map_index=map_index)
            .first()
        )
        if not ti:
            flash(f"Task [{dag_id}.{task_id}] does not exist", "error")
            return redirect(url_for("Airflow.index"))
        task_log_reader = TaskLogReader()
        if not task_log_reader.supports_external_link:
            flash("Task log handler does not support external links", "error")
            return redirect(url_for("Airflow.index"))
        handler = task_log_reader.log_handler
        url = handler.get_external_log_url(ti, try_number)
        return redirect(url)
    @expose("/task")
    @auth.has_access(