### 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)

_add_columns = _edit_columns = [
        "conn_id",
        "conn_type",
        "description",
        "host",
        "schema",
        "login",
        "password",
        "port",
        "extra",
    ]
    
    
    add_form = edit_form = ConnectionFormProxy
    add_template = "airflow/conn_create.html"
    edit_template = "airflow/conn_edit.html"
    add_widget = ConnectionFormWidget
    edit_widget = ConnectionFormWidget
    base_order = ("conn_id", "asc")
    def _iter_extra_field_names(self) -> Iterator[tuple[str, str]]:
        """Iterate through provider-backed connection fields.
        Note that this cannot be a property (including a cached property)
        because Flask-Appbuilder attempts to access all members on startup, and
        using a property would initialize the providers manager too eagerly.
        """
        return ((k, v.field_name) for k, v in ProvidersManager().connection_form_widgets.items())
    @property
    def add_columns(self) -> list[str]: