Modernize React Native examples

[frovere]

What changed

• migrate the example app from removed React Native core SafeAreaView to react-native-safe-area-context
• replace legacy touchables and the mapped ScrollView with Pressable and FlatList
• use explicit slot keys across OTP examples
• move repeating Reanimated caret values to useDerivedValue
• remove an unused example component and hoist static example metadata

Why

The example app used APIs and rendering patterns that React Doctor identified as deprecated or error-prone. These updates keep the public library API unchanged while making the examples compatible with current React Native and easier to maintain.

Documentation example pages import these source files directly, so they receive the same corrected snippets.

Validation

• yarn typecheck
• yarn lint
• yarn test --runInBand — 48 tests passed
• yarn workspace docs build
• npx react-doctor@latest --verbose — 100/100, no issues