### 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)

"clear",
        lazy_gettext("Clear"),
        lazy_gettext(
            "Are you sure you want to clear the state of the selected task"
            " instance(s) and set their dagruns to the QUEUED state?"
        ),
        single=False,
    )
    @action_has_dag_edit_access
    @provide_session
    @action_logging
    def action_clear(self, task_instances, session: Session = NEW_SESSION):
        """Clears the action."""
        try:
            dag_to_tis = collections.defaultdict(list)
            for ti in task_instances:
                dag = get_airflow_app().dag_bag.get_dag(ti.dag_id)
                dag_to_tis[dag].append(ti)
            for dag, task_instances_list in dag_to_tis.items():
                models.clear_task_instances(task_instances_list, session, dag=dag)
            session.commit()
            flash(f"{len(task_instances)} task instances have been cleared")
        except Exception as e:
            flash(f'Failed to clear task instances: "{e}"', "error")
        self.update_redirect()
        return redirect(self.get_redirect())
    @action("muldelete", "Delete", "Are you sure you want to delete selected records?", single=False)
    @action_has_dag_edit_access
    @action_logging
    def action_muldelete(self, items):