chore(data,engine,docs): remove Google Play Games Services sync backend

Deletes the solitaire_gpgs crate and all GPGS references from settings,
sync client, profile plugin, CLAUDE.md, and ARCHITECTURE.md. The
self-hosted server covers all sync needs without the Android-only backend.

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

solitaire_data/src/sync_client.rs

@@ -412,19 +412,12 @@ async fn extract_push_body(resp: reqwest::Response) -> Result<SyncResponse, Sync
 /// This is the **one** place in the codebase that matches on [`SyncBackend`]
 /// variants. All other code receives a `Box<dyn SyncProvider + Send + Sync>`
 /// and remains backend-agnostic.
-///
-/// `GooglePlayGames` is Android-only; on desktop it silently falls back to
-/// [`LocalOnlyProvider`].
 pub fn provider_for_backend(backend: &SyncBackend) -> Box<dyn SyncProvider + Send + Sync> {
     match backend {
         SyncBackend::Local => Box::new(LocalOnlyProvider),
         SyncBackend::SolitaireServer { url, username } => {
             Box::new(SolitaireServerClient::new(url.clone(), username.clone()))
         }
-        SyncBackend::GooglePlayGames => {
-            // GPGS is Android-only; fall back to no-op on desktop.
-            Box::new(LocalOnlyProvider)
-        }
     }
 }
 
@@ -470,12 +463,6 @@ mod tests {
         assert_eq!(provider.backend_name(), "local");
     }
 
-    #[test]
-    fn factory_gpgs_falls_back_to_local() {
-        let provider = provider_for_backend(&SyncBackend::GooglePlayGames);
-        assert_eq!(provider.backend_name(), "local");
-    }
-
     #[test]
     fn factory_server_returns_server_client() {
         let provider = provider_for_backend(&SyncBackend::SolitaireServer {