ci: Move lychee link check to script/check-links (#32460)

Follow-up to: https://github.com/zed-industries/zed/pull/30844

Transient link failure was blocking PR tests passing. [Example
run](https://github.com/zed-industries/zed/actions/runs/15560960788/job/43812878693?pr=32458).

Release Notes:

- N/A

.github/actions/build_docs/action.yml

@@ -19,12 +19,6 @@ runs:
       shell: bash -euxo pipefail {0}
       run: ./script/linux
 
-    - name: Check for broken links
-      uses: lycheeverse/lychee-action@82202e5e9c2f4ef1a55a3d02563e1cb6041e5332 # v2.4.1
-      with:
-        args: --no-progress './docs/src/**/*'
-        fail: true
-
     - name: Build book
       shell: bash -euxo pipefail {0}
       run: |