### 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
msg = "%s [%s]: " % (prompt, default)
            elif prompt:
                msg = "%s: " % prompt
            else:
                msg = 'input for %s: ' % varname
            if confirm:
                while True:
                    result = do_prompt(msg, private)
                    second = do_prompt("confirm " + msg, private)
                    if result == second:
                        break
                    self.display("***** VALUES ENTERED DO NOT MATCH ****")
            else:
                result = do_prompt(msg, private)
        else:
            result = None
            self.warning("Not prompting as we are not in interactive mode")
        
        if not result and default is not None:
            result = default
        if encrypt:
            
            from ansible.utils.encrypt import do_encrypt
            result = do_encrypt(result, encrypt, salt_size=salt_size, salt=salt)
        
        result = to_text(result, errors='surrogate_or_strict')
        if unsafe:
            result = wrap_var(result)
        return result
    def _set_column_width(self) -> None: