### 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._final_q: FinalQueue | None = None
        
        
        self._lock = threading.RLock()
        self.columns = None
        self.verbosity = verbosity
        if C.LOG_VERBOSITY is None:
            self.log_verbosity = verbosity
        else:
            self.log_verbosity = max(verbosity, C.LOG_VERBOSITY)
        
        self._deprecations: dict[str, int] = {}
        self._warns: dict[str, int] = {}
        self._errors: dict[str, int] = {}
        self.b_cowsay: bytes | None = None
        self.noncow = C.ANSIBLE_COW_SELECTION
        self.set_cowsay_info()
        if self.b_cowsay:
            try:
                cmd = subprocess.Popen([self.b_cowsay, "-l"], stdout=subprocess.PIPE, stderr=subprocess.PIPE)
                (out, err) = cmd.communicate()
                if cmd.returncode:
                    raise Exception
                self.cows_available: set[str] = {to_text(c) for c in out.split()}
                if C.ANSIBLE_COW_ACCEPTLIST and any(C.ANSIBLE_COW_ACCEPTLIST):
                    self.cows_available = set(C.ANSIBLE_COW_ACCEPTLIST).intersection(self.cows_available)
            except Exception:
                
                self.b_cowsay = None
        self._set_column_width()