refactor: remove unnecessary `lint-staged` operations (#65861)
### Why?
The current `lint-staged` hook uses [function
signature](https://github.com/lint-staged/lint-staged?tab=readme-ov-file#function-signature)
to target the **staged files** and has extra steps for handling those
files that seem to have been passed on from a [5-year-old legacy
code](https://github.com/vercel/next.js/pull/9245).
The entire process is unnecessary today since `lint-staged` only runs on
the staged files.
### How?
This PR removed unnecessary logic, removed some no-effect flags, and
[removed `git
add`](https://github.com/lint-staged/lint-staged?tab=readme-ov-file#v10).
> If your task previously contained a git add step, please remove this.
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Co-authored-by: Sam Ko <sam@vercel.com>
