feat: replace vanilla JS by AlpineJS

crates/app/templates/layout/nav/desktop/profile.html

@@ -1,5 +1,8 @@
 <!-- Profile dropdown -->
-<div class="relative ml-3">
+<div
+  class="relative ml-3"
+  x-data="{ profileOpen: false }"
+>
   <div>
     <button
       type="button"
@@ -8,6 +11,7 @@
       aria-controls="profile-dropdown"
       aria-expanded="false"
       aria-haspopup="menu"
+      x-on:click="profileOpen = ! profileOpen"
     >
       <span class="absolute -inset-1.5"></span>
       <span class="sr-only">Open user menu</span>