### 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)

for table_name in inspect(session.get_bind()).get_table_names():
                segments = table_name.split("__", 3)
                if len(segments) < 3:
                    continue
                if segments[0] != settings.AIRFLOW_MOVED_TABLE_PREFIX:
                    continue
                
                yield segments[-1], table_name
        if (
            permissions.ACTION_CAN_ACCESS_MENU,
            permissions.RESOURCE_ADMIN_MENU,
        ) in user_permissions and conf.getboolean("webserver", "warn_deployment_exposure"):
            robots_file_access_count = (
                session.query(Log)
                .filter(Log.event == "robots")
                .filter(Log.dttm > (utcnow() - datetime.timedelta(days=7)))
                .count()
            )
            if robots_file_access_count > 0:
                flash(
                    Markup(
                        "Recent requests have been made to /robots.txt. "