[State Sync Redesign 3.11] Update state logs and debug tools

"Download State Logs" in Settings is a user-facing debug tool used by support teams. The exported format must remain usable.

Current consumers:
- `getCleanAppState(store)` returns `{ ...store.getState(), metamask: { ...state.metamask, socialLoginEmail: undefined } }`
- `getLogs` reads `reduxState.metamask.logs`
- `logStateString` (bound to `window.logStateString`) calls both
- E2E tests read state via `window.stateHooks.getCleanAppState()`
- E2E `errors.spec.ts` asserts `extra.appState.state.metamask.participateInMetaMetrics`

## Steps

1. Update `getCleanAppState` to build state from `ControllerStateManager` snapshots (controller-keyed output)
2. Update `getLogs` to read from `ControllerStateManager.getSnapshot('LoggingController').logs`
3. Update `global.metamask` API (`updateCurrentLocale`, `setFeatureFlag`)
4. Update E2E test assertions that read `state.metamask` from `getCleanAppState` — update to controller-keyed paths
5. Update E2E `errors.spec.ts` Sentry payload assertion
6. Update integration test preloaded state (`test/data/integration-init-state.json`, notification test mocks)
7. **Decision:** Should state logs output remain flat (add a flattening shim for backward compat with support workflows) or switch to controller-keyed? If flat, add `ControllerStateManager.getFlatSnapshot()` utility.

## Files

- `ui/index.js`
- `test/e2e/helpers.js`
- `test/e2e/tests/metrics/errors.spec.ts`
- `test/data/integration-init-state.json`
- `test/integration/notifications&auth/`

## Risk

**Medium** — changing state logs format may affect support workflows. E2E test updates are mechanical but numerous.

## Dependencies

Issue 3.10 (Sentry update)

## Context

Part of [State Sync Redesign — Epic 3: UI-controller subscriptions].

Start by tracing ControllerStateManager usage across ui/index.js and test/e2e/helpers.js, then inspect the metrics errors test and integration state fixtures listed in the issue. Determine whether support workflows require flat logs or can use controller-keyed output, then update the APIs, assertions, mocks, and snapshots consistently and run the affected E2E and integration tests.

Written by the indexing model from the issue text.