[iOS] Now Playing audio session not visible on Apple Watch despite working on Lock Screen

Description
When playing audio/video in Brave iOS (e.g. YouTube Music via browser), media controls appear correctly on the iPhone Lock Screen, confirming MPNowPlayingInfoCenter is being set. However, the audio session does not appear in Apple Watch's Now Playing
complication or app — making it impossible to control Brave playback from the watch.

Steps to Reproduce
1. Open YouTube Music (or any media) in Brave iOS
2. Play audio, lock the phone
3. Confirm playback controls appear on Lock Screen ✅
4. Check Apple Watch → Now Playing → Brave session does not appear ❌

Expected Behavior
Brave's audio session should appear in Apple Watch Now Playing, since it already registers with MPNowPlayingInfoCenter (proven by Lock Screen controls working).

Likely Cause
The AVAudioSession category may not be set to .playback with proper activation, or MPNowPlayingSession.becomeActiveIfPossible() isn't being called — which would prevent the session from propagating to connected devices (Watch, AirPods remote, etc.).
This is especially relevant for Brave Playlist, which is marketed as a media playback feature.

Environment
- Brave iOS (latest)
- iPhone + Apple Watch

Labels: OS/iOS, feature/playlist

Start by tracing Brave iOS audio playback through AVAudioSession, MPNowPlayingInfoCenter, and MPNowPlayingSession.becomeActiveIfPossible(). Reproduce with YouTube Music or another media source while checking the iPhone Lock Screen and Apple Watch Now Playing. Done means the active Brave session appears on the watch and remains controllable there.

Written by the indexing model from the issue text.