### 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)

"""Set state to 'failed'."""
        self.set_task_instance_state(tis, State.FAILED)
        self.update_redirect()
        return redirect(self.get_redirect())
    @action("set_success", "Set state to 'success'", "", single=False)
    @action_has_dag_edit_access
    @action_logging
    def action_set_success(self, tis):
        """Set state to 'success'."""
        self.set_task_instance_state(tis, State.SUCCESS)
        self.update_redirect()
        return redirect(self.get_redirect())
    @action("set_retry", "Set state to 'up_for_retry'", "", single=False)
    @action_has_dag_edit_access
    @action_logging
    def action_set_retry(self, tis):
        """Set state to 'up_for_retry'."""
        self.set_task_instance_state(tis, State.UP_FOR_RETRY)
        self.update_redirect()
        return redirect(self.get_redirect())
    @action("set_skipped", "Set state to 'skipped'", "", single=False)
    @action_has_dag_edit_access
    @action_logging
    def action_set_skipped(self, tis):
        """Set state to skipped."""
        self.set_task_instance_state(tis, TaskInstanceState.SKIPPED)
        self.update_redirect()
        return redirect(self.get_redirect())
class AutocompleteView(AirflowBaseView):