RN: Improve `lint-ci` Script (#51853)

Summary:
Pull Request resolved: https://github.com/facebook/react-native/pull/51853

While testing D75988059 with D76047973, I noticed a few opportunities to improve the script used by `yarn lint-ci`:

- The exit code does not currently propagate, meaning `lint-ci` will succeed when it shouldn't.
- The shell script uses some non-idiomatic practices, so this improves it.

Changelog:
[Internal]

Reviewed By: kassens, cipolleschi

Differential Revision: D76049502

fbshipit-source-id: ebefddf0909edc89cd97de756b93bfaaa3d7cdef