### 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)

"""Form widget used to display connection."""
    @cached_property
    def field_behaviours(self) -> str:
        return json.dumps(ProvidersManager().field_behaviours)
    @cached_property
    def testable_connection_types(self) -> list[str]:
        return [
            connection_type
            for connection_type, hook_info in ProvidersManager().hooks.items()
            if hook_info and hook_info.connection_testable
        ]
class ConnectionFormProxy:
    """A stand-in for the connection form class.
    Flask-Appbuilder model views only ever call the ``refresh()`` function on
    the form class, so this is the perfect place to make the form generation
    dynamic. See docstring of ``create_connection_form_class`` for rationales.
    """
    @staticmethod
    def refresh(obj=None):
        return create_connection_form_class().refresh(obj)
class ConnectionModelView(AirflowModelView):