fix(engine): freeze game timers while the Home picker is up

The HUD's elapsed-time counter ticked from the moment the default
Classic deal landed at startup, even though the auto-show Home
picker was still up — so the player saw "0:11" before they had
chosen a mode. Time Attack had the same issue when M was pressed
mid-session: the 10-minute countdown burned while the player browsed
modes.

`tick_elapsed_time` and `advance_time_attack` now also gate on the
absence of `HomeScreen`, mirroring their existing `PausedResource`
check. The Home modal already covers input via its scrim, so this
purely freezes the timer without coupling to the pause-overlay
ownership of `PausedResource`.

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

solitaire_engine/src/game_plugin.rs

@@ -252,16 +252,20 @@ pub fn advance_elapsed(
 }
 
 /// Increment `GameState.elapsed_seconds` once per real-world second while
-/// the game is in progress (not won) and not paused. Stops counting on
+/// the game is in progress (not won), not paused, and the launch /
-/// win so the final time reflects how long the player took to solve the
+/// mode-picker Home modal isn't covering the board. Stops counting on
-/// deal; stops while the pause overlay is open.
+/// win so the final time reflects how long the player took to solve
+/// the deal; stops while the pause overlay is open; stops while Home
+/// is up so the timer doesn't tick under the picker before the player
+/// has actually committed to a deal.
 fn tick_elapsed_time(
     time: Res<Time>,
     mut game: ResMut<GameStateResource>,
     mut accumulator: Local<f32>,
     paused: Option<Res<crate::pause_plugin::PausedResource>>,
+    home_screens: Query<(), With<crate::home_plugin::HomeScreen>>,
 ) {
-    if paused.is_some_and(|p| p.0) {
+    if paused.is_some_and(|p| p.0) || !home_screens.is_empty() {
         return;
     }
     let is_won = game.0.is_won;

solitaire_engine/src/time_attack_plugin.rs

@@ -171,11 +171,15 @@ fn advance_time_attack(
     mut ended: MessageWriter<TimeAttackEndedEvent>,
     paused: Option<Res<crate::pause_plugin::PausedResource>>,
     path: Option<Res<TimeAttackSessionPath>>,
+    home_screens: Query<(), With<crate::home_plugin::HomeScreen>>,
 ) {
     if !session.active {
         return;
     }
-    if paused.is_some_and(|p| p.0) {
+    // Mirrors `tick_elapsed_time`: pause while the launch / mode-picker
+    // Home modal is up so the countdown doesn't burn while the player
+    // is choosing what to play next.
+    if paused.is_some_and(|p| p.0) || !home_screens.is_empty() {
         return;
     }
     session.remaining_secs -= time.delta_secs();