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[RN] Update ReactPrivate and InitializeCore imports (0.87 Strict API) (#36986)

## Summary

Migrate internal `Libraries/` import subpaths from `react-native` to the
incoming dedicated entry points in React Native 0.87.

**Motivation**

See [**RFC0894: Removing deep imports from
react-native**](https://github.com/react-native-community/discussions-and-proposals/pull/894)

**Changes**

The `react-private-interface` entry point has been created explicitly
for the private contract between React ↔ React Native, and is 1:1 with
the previous `ReactNativePrivateInterface` module.

Both of the below `Libraries/` paths still exist, but are deprecated —
this is a lagging migration that will enable cleanup in a future RN
version.

- `react-native/Libraries/Core/InitializeCore` →
`react-native/setup-env`
[react/react-native#57475](https://github.com/react/react-native/pull/57475)
- `react-native/Libraries/ReactPrivate/ReactNativePrivateInterface` →
`react-native/react-private-interface`
[react/react-native#57495](https://github.com/react/react-native/pull/57495)

## Test Plan

Flow
A
Alex Hunt committed
96fcba90138e9f1a73ee1cc4f79a653ec12fc3a9
Parent: 9ceb1e7
Committed by GitHub <noreply@github.com> on 7/28/2026, 3:45:37 PM