fix: align youtube playback with shared overlay startup

src/anki-integration/runtime.ts

@@ -9,6 +9,7 @@ import {
 export interface AnkiIntegrationRuntimeProxyServer {
   start(options: { host: string; port: number; upstreamUrl: string }): void;
   stop(): void;
+  waitUntilReady(): Promise<void>;
 }
 
 interface AnkiIntegrationRuntimeDeps {
@@ -131,6 +132,13 @@ export class AnkiIntegrationRuntime {
     return this.config;
   }
 
+  waitUntilReady(): Promise<void> {
+    if (!this.started || !this.isProxyTransportEnabled()) {
+      return Promise.resolve();
+    }
+    return this.getOrCreateProxyServer().waitUntilReady();
+  }
+
   start(): void {
     if (this.started) {
       this.stop();