### 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 ModelView prevents ability to pass primary keys of objects relating to DAGs you shouldn't be able to
    edit. This only holds for the add, update and delete operations.
    You will still need to use the `action_has_dag_edit_access()` for actions.
    """
    @staticmethod
    def validate_dag_edit_access(item: DagRun | TaskInstance):
        """Validates whether the user has 'can_edit' access for this specific DAG."""
        if not get_airflow_app().appbuilder.sm.can_edit_dag(item.dag_id):
            raise AirflowException(f"Access denied for dag_id {item.dag_id}")
    def pre_add(self, item: DagRun | TaskInstance):
        self.validate_dag_edit_access(item)
    def pre_update(self, item: DagRun | TaskInstance):
        self.validate_dag_edit_access(item)
    def pre_delete(self, item: DagRun | TaskInstance):
        self.validate_dag_edit_access(item)
    def post_add_redirect(self):  
        return redirect(self.get_default_url())
    def post_edit_redirect(self):  
        return redirect(self.get_default_url())
    def post_delete_redirect(self):