fix(chat): no chat field or nudge on a finished game

A finished game is read-only: gate the nudge behind canNudge (active game,
opponent's turn) so it no longer shows on a finished (or otherwise
non-active) game — where the backend rejects it anyway. The message field
is already hidden off your own turn. Extend the finished-game e2e to assert
neither Send nor Nudge is offered.

ui/e2e/social.spec.ts

@@ -116,6 +116,9 @@ test('finished game draws an inert footer and trims live-only controls', async (
   await page.getByRole('button', { name: 'Chat' }).click(); // 💬
   await expect(page.locator('.pane')).toHaveCount(1);
   await expect(page.getByRole('button', { name: 'Dictionary' })).toHaveCount(0);
+  // A finished game is read-only: neither the message field nor the nudge is offered.
+  await expect(page.getByRole('button', { name: 'Send' })).toHaveCount(0);
+  await expect(page.getByRole('button', { name: 'Nudge' })).toHaveCount(0);
 });
 
 test('lobby: hiding a finished game removes it (kebab → ❌), keeping the others', async ({ page }) => {