fix(example): preserve concurrent message and pagination updates

[OskarEichler]

Fixes

Use functional state for appending/prepending messages and pages, compute new chat messages from current rather than captured lengths, and label the unstable-chat scroll control with its actual target index.

Compatibility / observable changes

Example-only changes. Batched updates no longer overwrite each other and added messages use the current alternating sender. No library API change.

Verification

• Consumer verification: the combined fixes were packaged on the unchanged npm 2.3.2 runtime baseline and checked with React Native 0.87.1 / React 19.2.3. Immutable install, lint, both Android Debug variants, both unsigned iOS Simulator variants, four Metro bundles, 13 web targets and four browser-extension builds pass. All 298 installed non-metadata package files match the reviewed artifact byte-for-byte.
• Independent patch applied to 435b5141df4960ebbbc0b93264db3f5a4b23fffd: all 187 existing tests (14 suites), library TypeScript build and changed-file ESLint pass.
• One-off actual-source regression diagnostics pass:
  • fixture/react-native/src/Chat appends using current length
  • fixture/web/Chat appends using current length
  • Messages preserves batched messages
  • MessagesFlatList preserves batched messages
  • pagination preserves concurrent pages
• The combined review workspace passes Android Debug, unsigned iOS Simulator Debug/Release, both production Metro bundles, web production export, Docusaurus/Jekyll builds and all 24 existing iOS Detox tests (10 suites). The contacts comparison was rerun after its final change. These combined-workspace results are not a claim that every device/platform was tested independently for this PR.
• No checked-in tests, reference screenshots, dependency versions or native SDK versions were changed.

Reviewers’ hat-rack 🎩

• Check the stated compatibility behavior and reproduce the relevant cases above.
• Run the existing test/type/lint commands and exercise the affected example or list behavior on your supported device matrix.

Physical-device behavior, Android Detox, assistive-technology conformance and release signing were not verified. No app deployment is part of this PR.

[OskarEichler]

I have signed the CLA!