### 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 u": [" in msg:
            msg = msg.replace(u"[", u"")
            if msg.endswith(u"]"):
                msg = msg[:-1]
        runcmd = [self.b_cowsay, b"-W", b"60"]
        if self.noncow:
            thecow = self.noncow
            if thecow == 'random':
                thecow = random.choice(list(self.cows_available))
            runcmd.append(b'-f')
            runcmd.append(to_bytes(thecow))
        runcmd.append(to_bytes(msg))
        cmd = subprocess.Popen(runcmd, stdout=subprocess.PIPE, stderr=subprocess.PIPE)
        (out, err) = cmd.communicate()
        self.display(u"%s\n" % to_text(out), color=color)
    def error(self, msg: str, wrap_text: bool = True) -> None:
        if wrap_text:
            new_msg = u"\n[ERROR]: %s" % msg
            wrapped = textwrap.wrap(new_msg, self.columns)
            new_msg = u"\n".join(wrapped) + u"\n"
        else:
            new_msg = u"ERROR! %s" % msg
        if new_msg not in self._errors:
            self.display(new_msg, color=C.COLOR_ERROR, stderr=True)
            self._errors[new_msg] = 1
    @staticmethod
    def prompt(msg: str, private: bool = False) -> str: