Remove --no-stash flag from lint-staged in pre-commit hook (#6188)
### Problem
The `--no-stash` flag was originally added to work around issues where teammates hadn't installed Biome after our migration from Prettier. When the formatter failed, lint-staged's default stashing behavior would remove staged changes, causing frustration.
However, this workaround now causes a different problem: lint-staged runs formatters on files containing both staged AND unstaged changes. When Biome formats these files, it inadvertently stages unstaged modifications, effectively merging work-in-progress changes into commits.
### Solution
Remove the `--no-stash` flag to restore lint-staged's default behavior:
- Stash unstaged changes before running formatters
- Run formatters only on staged content
- Restore the stash after formatting
This ensures that only intentionally staged changes are included in commits, preventing accidental inclusion of work-in-progress modifications.
### Context
It's been sufficient time since the Prettier → Biome migration that all team members should have the proper tooling installed. The original workaround is no longer needed and is now causing more problems than it solves.
Bee