fix: server-side elapsed tracking via frame count, periodic sync, playlist respects play mode

Progress/lyrics sync:
- AudioPlayer tracks framesPlayed (ground truth: frames * 20ms + seekOffset)
- BotStatus.elapsed is the real elapsed from server
- Frontend polls GET /elapsed every 3s for ground truth
- Client interpolates between syncs (serverElapsed + timeSinceSync)
- Pause freezes at interpolated value, resume resyncs from server
- No more client-side timestamp drift

Playlist play all:
- Stops current playback before loading
- Respects play mode: random/rloop picks random first song, seq plays first

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

src/bot/instance.ts

@@ -37,8 +37,7 @@ export interface BotStatus {
   queueSize: number;
   volume: number;
   playMode: PlayMode;
-  seekOffset: number; // seconds offset from last seek/play start
-  playStartTime: number; // server timestamp when playback started
+  elapsed: number; // ground truth elapsed seconds from frame count
 }
 
 export class BotInstance extends EventEmitter {
@@ -507,8 +506,7 @@ export class BotInstance extends EventEmitter {
       queueSize: this.queue.size(),
       volume: this.player.getVolume(),
       playMode: this.queue.getMode(),
-      seekOffset: this.player.getSeekOffset(),
-      playStartTime: Date.now(),
+      elapsed: this.player.getElapsed(),
     };
   }