Refactor AppDelegate (milestone 1) (#3727)

Task/Issue URL: https://app.asana.com/0/0/1208832732122403/f
Tech Design URL: https://app.asana.com/0/481882893211075/1208859623176995/f
CC: @bwaresiak 

**Description**:
- Migrate the app’s startup flow into a state machine with clearly
defined states, each specifying its dependencies to ensure the necessary
resources are available before execution.
- Keep the flow primarily synchronous as it currently is, with some
exceptions for existing async code.
- Define explicit state dependencies to avoid optional objects, ensuring
clarity about what exists in each state.

Core/Pixel.swift

@@ -165,16 +165,6 @@ public struct PixelParameters {
 
     public static let appState = "state"
     public static let appEvent = "event"
-
-    public static let firstBackgroundTimestamp = "firstBackgroundTimestamp"
-    public static let secondBackgroundTimestamp = "secondBackgroundTimestamp"
-    public static let didReceiveMemoryWarningTimestamp = "didReceiveMemoryWarningTimestamp"
-    public static let didReceiveMXPayloadTimestamp = "didReceiveMXPayloadTimestamp"
-    public static let didReceiveUNNotification = "didReceiveUNNotification"
-    public static let didStartRemoteMessagingClientBackgroundTask = "didStartRemoteMessagingClientBackgroundTask"
-    public static let didStartAppConfigurationFetchBackgroundTask = "didStartAppConfigurationFetchBackgroundTask"
-    public static let didPerformFetchTimestamp = "didPerformFetchTimestamp"
-    public static let numberOfBackgrounds = "numberOfBackgrounds"
 }
 
 public struct PixelValues {

Core/PixelEvent.swift

@@ -971,7 +971,6 @@ extension Pixel {
 
         // MARK: Lifecycle
         case appDidTransitionToUnexpectedState
-        case appDidConsecutivelyBackground
     }
 
 }
@@ -1939,8 +1938,7 @@ extension Pixel.Event {
         case .openAIChatFromAddressBar: return "m_aichat_addressbar_icon"
 
         // MARK: Lifecycle
-        case .appDidTransitionToUnexpectedState: return "m_debug_app-did-transition-to-unexpected-state"
-        case .appDidConsecutivelyBackground: return "m_debug_app-did-consecutively-background"
+        case .appDidTransitionToUnexpectedState: return "m_debug_app-did-transition-to-unexpected-state-2"
 
         }
     }

DuckDuckGo/AppConfigurationFetch.swift

@@ -85,8 +85,6 @@ class AppConfigurationFetch {
         return Date().timeIntervalSince(Self.lastConfigurationRefreshDate) > Constants.minimumConfigurationRefreshInterval
     }
 
-    static var didStartBackgroundTaskTimestamp: Date?
-
     enum BackgroundRefreshCompletionStatus {
 
         case expired
@@ -134,7 +132,6 @@ class AppConfigurationFetch {
 
     static func registerBackgroundRefreshTaskHandler() {
         BGTaskScheduler.shared.register(forTaskWithIdentifier: Constants.backgroundProcessingTaskIdentifier, using: nil) { (task) in
-            didStartBackgroundTaskTimestamp = Date()
             guard shouldRefresh else {
                 task.setTaskCompleted(success: true)
                 scheduleBackgroundRefreshTask()

DuckDuckGo/AppDelegate+AppDeepLinks.swift

@@ -20,7 +20,7 @@
 import UIKit
 import Core
 
-extension AppDelegate {
+extension OldAppDelegate {
 
     func handleAppDeepLink(_ app: UIApplication, _ mainViewController: MainViewController?, _ url: URL) -> Bool {
         guard let mainViewController else { return false }