Use new mailto handling setup API.

Re-lands https://chromium-review.googlesource.com/1033019 after
fixes to downstream changes.

Bug: 810904
Cq-Include-Trybots: master.tryserver.chromium.mac:ios-simulator-cronet;master.tryserver.chromium.mac:ios-simulator-full-configs
Change-Id: I0b168ae145999d3269d910b77c6a770d86fb8ee0
Reviewed-on: https://chromium-review.googlesource.com/1034028Reviewed-by: Peter Lee <pkl@chromium.org>
Commit-Queue: Daniel Blakemore <danblakemore@google.com>
Cr-Commit-Position: refs/heads/master@{#554818}

ios/chrome/app/main_controller.mm

@@ -1126,24 +1126,19 @@ void MainControllerAuthenticationServiceDelegate::ClearBrowsingData(
 
 - (void)initializeMailtoHandling {
   if (base::FeatureList::IsEnabled(kMailtoHandledWithGoogleUI)) {
+    __weak __typeof(self) weakSelf = self;
     [[DeferredInitializationRunner sharedInstance]
         enqueueBlockNamed:kMailtoHandlingInitialization
                     block:^{
+                      __strong __typeof(weakSelf) strongSelf = weakSelf;
+                      if (!strongSelf) {
+                        return;
+                      }
                       MailtoHandlerProvider* provider =
                           ios::GetChromeBrowserProvider()
                               ->GetMailtoHandlerProvider();
-                      ios::ChromeIdentityService* identityService =
-                          ios::GetChromeBrowserProvider()
-                              ->GetChromeIdentityService();
                       provider->PrepareMailtoHandling(
-                          ^ChromeIdentity* {
-                            // TODO:(crbug.com/810904) Replace with currently
-                            // signed-in user.
-                            return nil;
-                          },
-                          ^NSArray<ChromeIdentity*>* {
-                            return identityService->GetAllIdentities();
-                          });
+                          strongSelf->_mainBrowserState);
                     }];
   }
 }