docs: Run lychee link check on generated docs output (#35381)

Closes #ISSUE

Following #35310, . This PR makes it so the lychee link check is ran
before building the docs on the md files to catch basic errors, and then
after building on the html output to catch generation errors, including
regressions like the one #35380 fixes.

Release Notes:

- N/A

.github/actions/build_docs/action.yml

@@ -19,7 +19,7 @@ runs:
       shell: bash -euxo pipefail {0}
       run: ./script/linux
 
-    - name: Check for broken links
+    - name: Check for broken links (in MD)
       uses: lycheeverse/lychee-action@82202e5e9c2f4ef1a55a3d02563e1cb6041e5332 # v2.4.1
       with:
         args: --no-progress --exclude '^http' './docs/src/**/*'
@@ -30,3 +30,9 @@ runs:
       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