…187269)

## Summary

Adds a Flutter Driver integration test under
`dev/integration_tests/android_engine_test/` covering `SystemUiMode`
transitions on Android. The test exposes a `requestData` handler that
applies modes via `SystemChrome.setEnabledSystemUIMode` and reads the
resulting decor-view `systemUiVisibility` flags via a new
`get_system_ui_visibility` method on the existing `native_driver` method
channel.

Closes flutter#187265.

## Why

flutter#186723 was a P1 regression introduced by flutter#183072 and fixed by flutter#187207.
The Mockito-based `PlatformPluginTest` unit tests verify which APIs are
called on a mocked `Window`/`View` but cannot observe residual
`decorView` state that bleeds across mode switches — which is the exact
bug class the regression was. A device-level integration test closes
that gap.

## What the tests cover

| Transition | Assertion |
|------------|-----------|
| `immersiveSticky → edgeToEdge` (regression case for flutter#186723) |
`FLAG_FULLSCREEN` and `FLAG_HIDE_NAVIGATION` are cleared |
| `immersive → edgeToEdge` | Same |
| `leanBack → edgeToEdge` | Same |
| `edgeToEdge → immersive` | `IMMERSIVE \| FULLSCREEN \|
HIDE_NAVIGATION` set |
| `edgeToEdge → immersiveSticky` | `IMMERSIVE_STICKY \| FULLSCREEN \|
HIDE_NAVIGATION` set |
| `edgeToEdge → leanBack` | `FULLSCREEN \| HIDE_NAVIGATION` set, no
immersive flags |

All tests skip on API < 29 since `edgeToEdge` requires Android 10+.

## Files

- New: `lib/system_ui_mode_transitions_main.dart` — Flutter Driver
entrypoint with mode-applying handler.
- New: `test_driver/system_ui_mode_transitions_main_test.dart` —
assertion suite.
- Modified: `NativeDriverSupportPlugin.kt` — adds
`get_system_ui_visibility` method that returns the decor view's
`systemUiVisibility` int. Reading the deprecated field is the most
direct signal of the regression class (flutter#186723 was specifically about
flag bleed-through on this field).
- Modified: `README.md` — adds the new entrypoint to the test list.

CI auto-discovery is via
`Glob('dev/integration_tests/android_engine_test/lib/**_main.dart')` in
`dev/bots/suite_runners/run_android_engine_tests.dart` — no separate
registration needed.

## Pre-launch Checklist

- [x] I read the [Contributor
Guide](https://github.com/flutter/flutter/blob/main/docs/contributing/Tree-hygiene.md#overview).
- [x] I read the [Tree
Hygiene](https://github.com/flutter/flutter/blob/main/docs/contributing/Tree-hygiene.md)
wiki page.
- [x] I read and followed the [Flutter Style
Guide](https://github.com/flutter/flutter/blob/main/docs/contributing/Style-guide-for-Flutter-repo.md).
- [x] I signed the [CLA](https://cla.developers.google.com/).
- [x] I listed at least one issue that this PR fixes in the description
above.
- [x] I added new tests to check the change I am making.
- [x] All existing and new tests are passing.

## Notes for reviewers

- Marked as draft because I do not have a local Android emulator set up
— `flutter analyze` is clean but I have not run the test end-to-end. CI
is the source of truth.
- The reason for reading the deprecated `systemUiVisibility` field on
the native side (rather than `WindowInsetsCompat.isVisible(...)`) is
that the regression was specifically about bit flags persisting on the
decor view; the deprecated read gives a deterministic,
version-independent signal. The production code still migrates to
non-deprecated APIs for *setting* state per flutter#183072.
- `SystemUiMode.manual` is out of scope here (different API surface —
takes a `List<SystemUiOverlay>`) and can follow up.