Alpine.js is not loaded in the Vue SPA bundle, causing the avatar dropdown
to never open and making Settings and Log out inaccessible. Replaced x-data/
x-show/x-transition/@click.away with Vue refs, onMounted/onUnmounted click-
outside listener, and Vue's built-in <Transition> component.
Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>