feat(engine): auto-show Home / mode picker on launch

The Home (mode picker) was only reachable via M during gameplay, so
players who hadn't discovered the hotkey never saw the Daily / Zen /
Challenge / Time Attack entry points after the splash cleared.

- HomePlugin gains an `auto_show_on_launch` flag (default true) and a
  matching `headless()` test constructor that disables it.
- spawn_home_on_launch flips a one-shot LaunchHomeShown flag once the
  splash has cleared, gated on RestorePromptScreen / PendingRestoredGame
  so the Welcome-back flow still takes precedence on machines with a
  saved game.
- App entry uses HomePlugin::default(); both headless test fixtures
  switch to HomePlugin::headless() so per-test worlds start clean.

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

solitaire_app/src/main.rs

@@ -129,7 +129,7 @@ fn main() {
         .add_plugins(TimeAttackPlugin)
         .add_plugins(HudPlugin)
         .add_plugins(HelpPlugin)
-        .add_plugins(HomePlugin)
+        .add_plugins(HomePlugin::default())
         .add_plugins(ProfilePlugin)
         .add_plugins(PausePlugin)
         .add_plugins(SettingsPlugin::default())