Staged docs releases (#50136)

Closes #ISSUE

Before you mark this PR as ready for review, make sure that you have:
- [x] Added a solid test coverage and/or screenshots from doing manual
testing
- [x] Done a self-review taking into account security and performance
aspects
- [x] Aligned any UI changes with the [UI
checklist](https://github.com/zed-industries/zed/blob/main/CONTRIBUTING.md#uiux-checklist)

Release Notes:

- N/A *or* Added/Fixed/Improved ...

.github/actions/build_docs/action.yml

@@ -1,46 +0,0 @@
-name: "Build docs"
-description: "Build the docs"
-
-runs:
-  using: "composite"
-  steps:
-    - name: Setup mdBook
-      uses: peaceiris/actions-mdbook@ee69d230fe19748b7abf22df32acaa93833fad08 # v2
-      with:
-        mdbook-version: "0.4.37"
-
-    - name: Cache dependencies
-      uses: swatinem/rust-cache@9d47c6ad4b02e050fd481d890b2ea34778fd09d6 # v2
-      with:
-        save-if: ${{ github.ref == 'refs/heads/main' }}
-        # cache-provider: "buildjet"
-
-    - name: Install Linux dependencies
-      shell: bash -euxo pipefail {0}
-      run: ./script/linux
-
-    - name: Download WASI SDK
-      shell: bash -euxo pipefail {0}
-      run: ./script/download-wasi-sdk
-
-    - name: Generate action metadata
-      shell: bash -euxo pipefail {0}
-      run: ./script/generate-action-metadata
-
-    - name: Check for broken links (in MD)
-      uses: lycheeverse/lychee-action@82202e5e9c2f4ef1a55a3d02563e1cb6041e5332 # v2.4.1
-      with:
-        args: --no-progress --exclude '^http' './docs/src/**/*'
-        fail: true
-
-    - name: Build book
-      shell: bash -euxo pipefail {0}
-      run: |
-        mkdir -p target/deploy
-        mdbook build ./docs --dest-dir=../target/deploy/docs/
-
-    - name: Check for broken links (in HTML)
-      uses: lycheeverse/lychee-action@82202e5e9c2f4ef1a55a3d02563e1cb6041e5332 # v2.4.1
-      with:
-        args: --no-progress --exclude '^http' 'target/deploy/docs/'
-        fail: true