### Summary

If you run `ANSIBLE_LOG_PATH=test ansible localhost -m setup`, `test` is now an empty file. Before #81692 got merged, it contained log output like
```
023-11-16 08:01:52,262 p=313108 u=felix n=ansible | [WARNING]: You are running the development version of Ansible. You should only run Ansible from "devel" if you are modifying the Ansible engine, or trying out features under development.
This is a rapidly changing source of code and can become unstable at any point.

2023-11-16 08:01:52,383 p=313108 u=felix n=ansible | [WARNING]: No inventory was parsed, only implicit localhost is available

2023-11-16 08:01:53,511 p=313108 u=felix n=ansible | localhost | SUCCESS => {
    "ansible_facts": {
...
```

### Issue Type

Bug Report

### Component Name

logging

### Ansible Version

```console
latest devel branch
```


### Configuration

```console
-
```


### OS / Environment

-

### Steps to Reproduce

Run `ANSIBLE_LOG_PATH=test ansible localhost -m setup`

### Expected Results

File `test` contains log output.

### Actual Results

```console
File `test` is empty.
```


### Code of Conduct

- [X] I agree to follow the Ansible Code of Conduct
self,
        msg: str,
        version: str | None = None,
        removed: bool = False,
        date: str | None = None,
        collection_name: str | None = None,
    ) -> None:
        if not removed and not C.DEPRECATION_WARNINGS:
            return
        message_text = self.get_deprecation_message(msg, version=version, removed=removed, date=date, collection_name=collection_name)
        if removed:
            raise AnsibleError(message_text)
        wrapped = textwrap.wrap(message_text, self.columns, drop_whitespace=False)
        message_text = "\n".join(wrapped) + "\n"
        if message_text not in self._deprecations:
            self.display(message_text.strip(), color=C.COLOR_DEPRECATE, stderr=True)
            self._deprecations[message_text] = 1
    @proxy_display
    def warning(self, msg: str, formatted: bool = False) -> None:
        if not formatted:
            new_msg = "[WARNING]: %s" % msg
            wrapped = textwrap.wrap(new_msg, self.columns)
            new_msg = "\n".join(wrapped) + "\n"
        else:
            new_msg = "\n[WARNING]: \n%s" % msg
        if new_msg not in self._warns:
            self.display(new_msg, color=C.COLOR_WARN, stderr=True)
            self._warns[new_msg] = 1
    def system_warning(self, msg: str) -> None: