### 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)

if get_airflow_app().appbuilder.sm.has_all_dags_access(g.user):
            return query
        filter_dag_ids = get_airflow_app().appbuilder.sm.get_accessible_dag_ids(g.user)
        return query.filter(self.model.dag_id.in_(filter_dag_ids))
class AirflowModelView(ModelView):
    """Airflow Mode View.
    Overridden `__getattribute__` to wraps REST methods with action_logger
    """
    list_widget = AirflowModelListWidget
    page_size = PAGE_SIZE
    CustomSQLAInterface = wwwutils.CustomSQLAInterface
    def __getattribute__(self, attr):
        """Wraps action REST methods with `action_logging` wrapper
        Overriding enables differentiating resource and generation of event name at the decorator level.
        if attr in ["show", "list", "read", "get", "get_list"]:
            return action_logging(event="RESOURCE_NAME"."action_name")(attr)
        else:
            return attr
        """
        attribute = object.__getattribute__(self, attr)
        if (
            callable(attribute)
            and hasattr(attribute, "_permission_name")
            and attribute._permission_name in self.method_permission_name
        ):
            permission_str = self.method_permission_name[attribute._permission_name]
            if permission_str not in ["show", "list", "read", "get", "get_list"]:
                return action_logging(event=f"{self.route_base.strip('/')}.{permission_str}")(attribute)
        return attribute
class AirflowPrivilegeVerifierModelView(AirflowModelView):