iOS: the icon's count follows the page's (#421)

Reported: the unread count on the icon only changed when the app was
opened. Two reasons:
- Items read in the page never reached it, since only the player's own
  mark-read refreshed the badge.
- The background refresh that should bring new items runs when iOS
  chooses, which for this app has been rarely.

The page now sends All Subscriptions' count ({t:"unread", n}) each time
its list is drawn with a new one, so reading, and new items arriving
over the event stream while the app is open, set it at once. While an
episode plays the app is awake with the screen locked, so the player's
tick also re-checks every quarter of an hour.

With the app closed and nothing playing, the count still waits on iOS's
refresh. Push (#433) is what makes that exact.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

tests/native-bridge.ts

@@ -175,6 +175,11 @@ since();
 vm.runInContext('toast("Unable to play", true)', ctx);
 ok(last('haptic')?.kind === 'warn', 'an error was not felt');
 
+// The icon's count (#421): the page's own, sent when it changes.
+since();
+vm.runInContext('hostBadge(7); hostBadge(7); hostBadge(3)', ctx);
+ok(posted.filter(m => m.t === 'unread').map(m => m.n).join() === '7,3', 'the app was not told the unread count once per change');
+
 if (failed) { console.error(`\n${failed} failed`); process.exit(1); }
 console.log('OK: native-bridge: the host takes playback, keeping, the sleep timer and haptics, and the page follows it');
 process.exit(0);