fix: select-bluetooth-device on Windows (#29612)

Co-authored-by: John Kleinschmidt <[email protected]>

shell/browser/lib/bluetooth_chooser.cc

@@ -55,6 +55,7 @@ void BluetoothChooser::SetAdapterPresence(AdapterPresence presence) {
       event_handler_.Run(content::BluetoothChooserEvent::CANCELLED, "");
       break;
     case AdapterPresence::POWERED_ON:
+      rescan_ = true;
       break;
   }
 }
@@ -87,7 +88,7 @@ void BluetoothChooser::ShowDiscoveryState(DiscoveryState state) {
     case DiscoveryState::DISCOVERING:
       // The first time this state fires is due to a rescan triggering so set a
       // flag to ignore devices
-      if (!refreshing_) {
+      if (rescan_ && !refreshing_) {
         refreshing_ = true;
       } else {
         // The second time this state fires we are now safe to pick a device

shell/browser/lib/bluetooth_chooser.h

@@ -42,6 +42,7 @@ class BluetoothChooser : public content::BluetoothChooser {
   EventHandler event_handler_;
   int num_retries_ = 0;
   bool refreshing_ = false;
+  bool rescan_ = false;
 
   DISALLOW_COPY_AND_ASSIGN(BluetoothChooser);
 };