Fix Site Switcher screen not dismissing after adding self-hosted site

RELEASE-NOTES.txt

@@ -1,5 +1,6 @@
 25.9
 -----
+* [*] Site Switcher screen now properly dismisses after successfully adding a self-hosted site. [#24273]
 
 * [*] Updated site display logic in sidebar to prioritize showing the currently selected site on iPad. [#24277]
 

WordPress/Classes/ViewRelated/Blog/Site Picker/SiteSwitcherView.swift

@@ -4,6 +4,7 @@ import WordPressUI
 
 final class SiteSwitcherViewController: UIHostingController<SiteSwitcherView>, UIPopoverPresentationControllerDelegate {
     private let viewModel: BlogListViewModel
+    private var signinObserver: NSObjectProtocol?
 
     init(configuration: BlogListConfiguration = .defaultConfig,
          addSiteAction: @escaping ((AddSiteMenuViewModel.Selection) -> Void),
@@ -28,6 +29,14 @@ final class SiteSwitcherViewController: UIHostingController<SiteSwitcherView>, U
         if #available(iOS 17, *) {
             AppTips.SitePickerTip().invalidate(reason: .actionPerformed)
         }
+
+        // Add observer for sign-in completion notification
+        let notificationName = NSNotification.Name(WordPressAuthenticationManager.WPSigninDidFinishNotification)
+        signinObserver = NotificationCenter.default.addObserver(forName: notificationName, object: nil, queue: nil) { [weak self] _ in
+            DispatchQueue.main.async {
+                self?.presentingViewController?.dismiss(animated: true)
+            }
+        }
     }
 
     @objc private func buttonCloseTapped() {