fix: hide spent rescue to prevent StateError crash; log per-attempt starts

Expose engine.rescueUsed getter and surface it through GameViewState so
the result overlay can omit the watch-ad FilledButton after a rescue has
been consumed, preventing a second tap from calling useContinue /
addExtraMoves and hitting their StateError guard. Give-up is promoted to
FilledButton when rescue is unavailable for clear affordance.

Also emit stageStart / endlessStart analytics in restart() so every
attempt (not just the first) is bracketed by a matching start event.

Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>

test/game/engine/game_engine_test.dart

@@ -193,6 +193,21 @@ void main() {
     });
   });
 
+  group('rescueUsed getter', () {
+    test('rescueUsed flag is exposed and flips after a rescue', () {
+      final engine = GameEngine(
+        _stage(moveLimit: 1),
+        generator: _smallPool(1),
+      );
+      expect(engine.rescueUsed, isFalse);
+      engine.tryPlace(0, 0, 0);
+      expect(engine.phase, GamePhase.stuck);
+      expect(engine.rescueUsed, isFalse);
+      engine.addExtraMoves();
+      expect(engine.rescueUsed, isTrue);
+    });
+  });
+
   group('dead board and continue', () {
     StageConfig deadStage() {
       // Checkerboard: only monos fit, and the injected pool has none small