fix: update ci to use latest docs preprocessing

[anish-sahoo]

• fixes out of date docs.yml step which diverged from the current docs preprocessing
• updated ci step to now use the same mise commands, future-proofing it

[ask-bonk]

I'm Bonk, and I've done a quick review of your PR.

This PR extracts the docs preprocessing/staging logic into a reusable docs:stage mise task and updates the CI workflow to use it, fixing a divergence where CI was skipping link preprocessing and using a different staging approach than local development.

After reviewing the changes, I don't see any actionable issues. The refactor is clean, correctly DRYs up the staging logic between docs, docs:serve, and CI, and fixes the real bug where the deployed docs weren't getting the same preprocessing as local builds.

LGTM

github run