### 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)

"""Base View to set Airflow related properties."""
    from airflow import macros
    route_base = ""
    extra_args = {
        
        "macros": macros,
        "get_docs_url": get_docs_url,
    }
    if not conf.getboolean("core", "unit_test_mode"):
        executor, _ = ExecutorLoader.import_default_executor_cls()
        extra_args["sqlite_warning"] = settings.engine.dialect.name == "sqlite"
        if not executor.is_production:
            extra_args["production_executor_warning"] = executor.__name__
        extra_args["otel_on"] = conf.getboolean("metrics", "otel_on")
    line_chart_attr = {
        "legend.maxKeyLength": 200,
    }
    def render_template(self, *args, **kwargs):