### 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)

return self._set_notification_property(items, "email_sent", True)
    @action(
        "mulemailsentfalse",
        "Set email sent to false",
        "Are you sure you want to mark these SLA alerts as emails not sent yet?",
        single=False,
    )
    def action_mulemailsentfalse(self, items: list[SlaMiss]):
        return self._set_notification_property(items, "email_sent", False)
    @provide_session
    def _set_notification_property(
        self,
        items: list[SlaMiss],
        attr: str,
        new_value: bool,
        session: Session = NEW_SESSION,
    ):
        try:
            count = 0
            for sla in items:
                count += 1
                setattr(sla, attr, new_value)
                session.merge(sla)
            session.commit()
            flash(f"{count} SLAMisses had {attr} set to {new_value}.")
        except Exception as ex:
            flash(str(ex), "error")
            flash("Failed to set state", "error")
        self.update_redirect()
        return redirect(self.get_default_url())