refactor(web): remove client-side BYO API key feature

The BYO (Bring Your Own) API key configuration for LLM and image
generation will be re-implemented via Cloudflare Workers. Remove
the client-side implementation to prepare for that migration.

TTS (text-to-speech) BYO key support is intentionally preserved.

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

app/api/insert-beat/route.ts

@@ -22,7 +22,7 @@ export async function POST(req: Request) {
   }
 
   try {
-    const base = loadEngineConfig(req.headers);
+    const base = loadEngineConfig();
     // See StartRequest.clientTts — BYO clients synth in-browser, so drop server TTS.
     const config = body.clientTts === true ? { ...base, tts: undefined } : base;
     const result = await requestInsertBeat(config, body);