fix: Remove setting of process.exitCode that breaks Jest tests (#45562)

Summary:
In Node 20, the script to run unit tests in CI (`scripts/run-ci-javascript-tests.js`) will fail, even when all the Jest tests pass. This happens because one of the JS modules being tested is setting `process.exitCode` (see https://github.com/jestjs/jest/issues/9324#issuecomment-1808090455).

Changes:

- Modified the affected module to throw an exception when failing, instead of setting the exit code
- Adjusted the unit test for that module

## Changelog:

[General] [Fixed] - Remove setting of process.exitCode that breaks Jest tests

Pull Request resolved: https://github.com/facebook/react-native/pull/45562

Test Plan:
Before this change, running `node scripts/run-ci-javascript-tests.js` would fail with Node 20.
After this change, it succeeds.

Reviewed By: blakef

Differential Revision: D60033582

Pulled By: cipolleschi

fbshipit-source-id: 71b7f4495d414e719a9bd2d892bd1bc3045ddd5d