Bind the Stay Awake indicator to the idle service

The indicator polled omarchy-toggle-idle over a Process and re-ran it on
a timer to toggle, while the CLI called back into the shell over IPC to
apply and refresh the state it had just changed. Now the indicator binds
straight to the idle service's stayAwake property and flips it in
process. The CLI only touches the state file, which the service already
watches, so toggling from keybindings and scripts still reaches the
shell without any reentrant IPC.

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

test/shell.d/fixtures/indicator-contract/shell.qml

@@ -24,10 +24,21 @@ ShellRoot {
     }
   }
 
+  QtObject {
+    id: idleService
+    property bool stayAwake: false
+    readonly property bool idleEnabled: !stayAwake
+    function setIdleEnabled(value) {
+      stayAwake = !value
+    }
+  }
+
   QtObject {
     id: mockShell
     function firstPartyServiceFor(id) {
-      return id === "omarchy.notifications" ? notificationService : null
+      if (id === "omarchy.notifications") return notificationService
+      if (id === "omarchy.idle") return idleService
+      return null
     }
   }
 
@@ -151,7 +162,7 @@ ShellRoot {
         stayAwake.moduleName = "StayAwake"
         root.injectBar(stayAwake)
         stayAwake.triggerPress(Qt.LeftButton)
-        root.assertTrue(root.commandCount("omarchy-toggle-idle") === 1, "Stay Awake left click runs idle toggle command")
+        root.assertTrue(idleService.stayAwake === true, "Stay Awake left click toggles the idle service")
       }
 
       root.writeResult()