### 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)

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            }, 400
        with create_session() as session:
            if lstripped_orderby == "uri":
                if order_by[0] == "-":
                    order_by = (DatasetModel.uri.desc(),)
                else:
                    order_by = (DatasetModel.uri.asc(),)
            elif lstripped_orderby == "last_dataset_update":
                if order_by[0] == "-":
                    order_by = (
                        func.max(DatasetEvent.timestamp).desc(),
                        DatasetModel.uri.asc(),
                    )
                    if session.bind.dialect.name == "postgresql":
                        order_by = (order_by[0].nulls_last(), *order_by[1:])
                else:
                    order_by = (
                        func.max(DatasetEvent.timestamp).asc(),
                        DatasetModel.uri.desc(),
                    )
                    if session.bind.dialect.name == "postgresql":
                        order_by = (order_by[0].nulls_first(), *order_by[1:])
            count_query = session.query(func.count(DatasetModel.id))
            has_event_filters = bool(updated_before or updated_after)
            query = (
                session.query(
                    DatasetModel.id,
                    DatasetModel.uri,
                    func.max(DatasetEvent.timestamp).label("last_dataset_update"),