web: AuthGate consolidates three "not authenticated" cards

Create AuthGate component as the single source of truth for the
unauthenticated UI. Cold renders show a manual sign-in button (IV7);
mid-session expiry auto-redirects via useEffect (IV7); auth_error shows
the distinct error card with a "Try again" button and never auto-retries
(IV6). Swap the inline AuthError cards at index, projects, and
SavedSearchesSection call sites.