@protected annotation on methods isn't inherited

### Situation

Considering the following public API:

```dart
class Super {
/// This is the public API.
void foo() {
doFoo();
print('done');
}

/// Subclasses should not call this method directly; instead, call [foo]
@protected
@mustCallSuper
void doFoo() {
print('doing foo...');
}
}

class Sub extends Super {
@override
void doFoo() {
super.doFoo();
print('doing more foo');
}
}
```

### Problem

The problem here is that the developer of `Sub` has inadvertently made the `doFoo()` method public. The analyzer won't warn developers if they do the following in their project:

```dart
void main() {
Sub sub = Sub();
sub.doFoo();
}
```

### Possible solutions:

Developers don't typically want to open up the visibility of override methods, so one possible solution is just to always inherit the `@protected` annotation from super methods.

However, maybe it's a valid use case to make a protected method public when overridden? If we choose this to be the case, we could default to inheriting the `@protected` annotation -- and introduce a `@public` annotation that explicitly makes a method public again?

### Dart info

```terminal
$ dart info

If providing this information as part of reporting a bug, please review the information
below to ensure it only contains things you're comfortable posting publicly.

#### General info

- Dart 3.3.0-16.0.dev (dev) (Wed Oct 11 21:04:42 2023 -0700) on "macos_arm64"
- on macos / Version 13.6 (Build 22G120)
- locale is en

#### Process info

| Memory | CPU | Elapsed time | Command line |
| -----: | ---: | -----------: | ------------------------------------------------------------------------------- |
| 21 MB | 0.0% | 03-06:48:38 | dart devtools --machine --try-ports 10 --allow-embedding |
| 35 MB | 0.0% | 03:40:44 | dart devtools --no-launch-browser |
| 92 MB | 0.0% | 03-06:48:38 | dart language-server --protocol=lsp --client-id=VS-Code --client-version=3.74.0 |
```

Start by reproducing the analyzer behavior with the Super/Sub example and trace how @protected is handled on overridden methods. Review the 28-comment discussion for the intended visibility semantics; done requires an agreed behavior and analyzer coverage for overrides and direct calls.

Written by the indexing model from the issue text.