feat(home): clean shared top bar with live online counter

The homepage duplicated the online count and recent-user avatars (hero plus
community sidebar), and the public pages each kept their own chrome. Put one
AuthTopBar at the top of index, login and register — logo + hotel name plus a
live online indicator and the language/clock/theme controls — and drop the
duplicated hero rows.

src/components/shared/public-avatar-contract.test.ts

@@ -11,7 +11,7 @@ const compactCallSites = new Map<string, number>([
 	["src/app/(site)/leaderboard/page.tsx", 1],
 	["src/app/(site)/me/page.tsx", 1],
 	["src/app/(site)/messages/page.tsx", 2],
-	["src/app/(site)/page.tsx", 2],
+	["src/app/(site)/page.tsx", 1],
 	["src/app/(site)/radio/leaderboard/page.tsx", 1],
 	["src/app/(site)/radio/shouts/page.tsx", 1],
 	["src/app/(site)/rankings/page.tsx", 1],