### 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
if C.ANSIBLE_NOCOWS:
            return
        if C.ANSIBLE_COW_PATH:
            self.b_cowsay = C.ANSIBLE_COW_PATH
        else:
            for b_cow_path in b_COW_PATHS:
                if os.path.exists(b_cow_path):
                    self.b_cowsay = b_cow_path
    @proxy_display
    def display(
        self,
        msg: str,
        color: str | None = None,
        stderr: bool = False,
        screen_only: bool = False,
        log_only: bool = False,
        newline: bool = True,
    ) -> None:
        """ Display a message to the user
        Note: msg *must* be a unicode string to prevent UnicodeError tracebacks.
        """
        if not isinstance(msg, str):
            raise TypeError(f'Display message must be str, not: {msg.__class__.__name__}')
        nocolor = msg
        if not log_only:
            has_newline = msg.endswith(u'\n')
            if has_newline:
                msg2 = msg[:-1]