Restructure step 0: Connection extracted from the coordinator

Connection owns the socket lifecycle, connect/dial with epochs,
reconnect, the clock offset, diagnostics upload and a sealed ConnState
signal; the hello is built by an injectable builder, the resume token
stays private, and the reconnect delay and silence limit are parameters.
The coordinator maps ConnState to the same health texts and keeps the
reconnect restore and drop handling (step 1 replaces them).

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

apps/desktop/lib/src/sync/sync_client.dart

@@ -53,7 +53,21 @@ class SyncClient {
     HelloMedia? media,
     String hostKey = '',
     String resume = '',
-  }) async {
+  }) =>
+      connectWith(
+        wsUri: wsUri,
+        hello: HelloMessage(
+          proto: RelayConstants.protoVersion,
+          secret: secret,
+          user: HelloUser(name: name, avatar: avatar),
+          media: media ?? const HelloMedia(name: '', size: 0, duration: 0),
+          hostKey: hostKey,
+          resume: resume,
+        ),
+      );
+
+  /// Dials [wsUri], sends [hello] and waits for the welcome (10 s).
+  Future<WelcomeMessage> connectWith({required Uri wsUri, required HelloMessage hello}) async {
     await disconnect();
     clockSync.reset();
     this.wsUri = wsUri;
@@ -78,14 +92,6 @@ class SyncClient {
       },
       cancelOnError: false,
     );
-    final hello = HelloMessage(
-      proto: RelayConstants.protoVersion,
-      secret: secret,
-      user: HelloUser(name: name, avatar: avatar),
-      media: media ?? const HelloMedia(name: '', size: 0, duration: 0),
-      hostKey: hostKey,
-      resume: resume,
-    );
     ch.sink.add(jsonEncode(hello.toJson()));
     _pendingWelcome = welcomeCompleter;
     try {