Migrate fully to GitHub Actions for Pages deployment (#173)

Legacy deploy environments (including simply having something in
gh-pages!) is deprecated.

GitHub really wants us to use their deploy-pages system for this, so
switch to it.

.github/workflows/publish.yml

@@ -10,15 +10,18 @@ on:
   workflow_dispatch:
 
 jobs:
-  doc-deploy:
+  build:
     runs-on: ubuntu-latest
 
     steps:
-      - uses: actions/checkout@v3
+      - uses: actions/checkout@v4
         with:
           persist-credentials: false
           submodules: recursive
 
+      - name: Configure GitHub Pages
+        uses: actions/configure-pages@v5
+
       - uses: actions/setup-node@v3
         with:
           node-version: 20
@@ -30,9 +33,25 @@ jobs:
         # Build the site using VuePress
         run: npm run docs:build
 
-      - name: Deploy 🚀
-        uses: JamesIves/github-pages-[email protected]
+      - name: Upload GitHub Pages artifact
+        uses: actions/upload-pages-artifact@v3
         with:
-          branch: gh-pages # The branch the action should deploy to.
-          folder: docs/.vuepress/dist # The folder the action should deploy.
-          clean: true # Automatically remove deleted files from the deploy branch
+          path: "docs/.vuepress/dist"
+
+  # Deployment job
+  deploy:
+    # Sets permissions of the GITHUB_TOKEN to allow deployment to GitHub Pages
+    permissions:
+      contents: read
+      pages: write
+      id-token: write
+
+    environment:
+      name: github-pages
+      url: ${{ steps.deployment.outputs.page_url }}
+    runs-on: ubuntu-latest
+    needs: build
+    steps:
+      - name: Deploy to GitHub Pages
+        id: deployment
+        uses: actions/deploy-pages@v4