fix(docs): render the Copy page button on real docs pages and fix hydration 404s

The Copy page split button shipped in #739 never rendered: it was wired
into pages/[...slug].vue while all /docs/* routes are handled by the
more specific pages/docs/[...slug].vue, and it was referenced as
<CopyPageDropdown /> while the auto-import name is DocsCopyPageDropdown.

- Render DocsCopyPageDropdown in the title row of pages/docs/[...slug].vue
- Delete the root catch-all pages/[...slug].vue: it served no content
  (only /docs/** exists in the content collection) and its competing
  route caused client-side 404s after hydration on trailing-slash docs
  URLs
- Drop the redundant nested <article> that caused hydration mismatches
  on every docs page
- Fix doubled item descriptions in the copy dropdown (UDropdownMenu
  renders item.description natively)
- Remove dead layouts/default.vue + TheHeader/TheSidebar/TheTableOfContents
  (unused since the docs layout took over) and add the ⌘K hint to the
  active search button

Verified with a full static build + headless-browser checks: button in
the title row in light/dark, 4-item dropdown with correct .md/ChatGPT/
Claude targets, fuzzy section-level search with the AI-tip footer, and
no hydration 404s on slash/no-slash hard loads.

Author: binaryk

docs-v3/components/TheHeader.vue

@@ -30,12 +30,7 @@
       </button>
 
       <template #item-label="{ item }">
-        <div class="flex flex-col text-left">
-          <span class="text-sm font-medium text-gray-900 dark:text-white">{{ item.label }}</span>
-          <span v-if="item.description" class="text-xs text-gray-500 dark:text-gray-400">
-            {{ item.description }}
-          </span>
-        </div>
+        <span class="text-sm font-medium text-gray-900 dark:text-white">{{ item.label }}</span>
       </template>
     </UDropdownMenu>
   </div>

docs-v3/components/TheSidebar.vue

@@ -34,7 +34,12 @@
 
           <!-- Desktop search with label (docs only) -->
           <div v-if="isDocsPage" class="hidden lg:block">
-            <UContentSearchButton label="Search..." :collapsed="false" />
+            <UContentSearchButton
+              label="Search documentation…"
+              :kbds="['meta', 'K']"
+              :collapsed="false"
+              class="w-64 justify-start text-gray-500 dark:text-gray-400"
+            />
           </div>
 
           <a

docs-v3/components/TheTableOfContents.vue

@@ -1,17 +1,16 @@
 <template>
   <NuxtLayout name="docs">
     <div v-if="post">
-      <article>
-        <header class="mb-8">
+      <header class="mb-8">
+        <div class="flex flex-col-reverse sm:flex-row sm:items-start sm:justify-between gap-4">
           <h1 class="text-4xl font-bold text-gray-900 dark:text-white mb-4">
             {{ post.title }}
           </h1>
-        </header>
-        
-        <div class="prose-docs">
-          <ContentRenderer :value="post" />
+          <DocsCopyPageDropdown class="shrink-0 sm:mt-2" />
         </div>
-      </article>
+      </header>
+
+      <ContentRenderer :value="post" />
     </div>
 
     <div v-else>