Remove remaining calls to deprecated MessageLoop methods on Mac.

ios/chrome/browser/browsing_data/ios_chrome_browsing_data_remover.mm

 // Copyright 2015 The Chromium Authors. All rights reserved.
 // Use of this source code is governed by a BSD-style license that can be
 // found in the LICENSE file.
 
 #include "ios/chrome/browser/browsing_data/ios_chrome_browsing_data_remover.h"
 
 #include <map>
 #include <set>
 #include <string>
 
 #include "base/bind.h"
 #include "base/bind_helpers.h"
 #include "base/callback.h"
 #include "base/logging.h"
 #include "base/metrics/histogram_macros.h"
 #include "base/metrics/user_metrics.h"
+#include "base/threading/thread_task_runner_handle.h"
 #include "components/autofill/core/browser/personal_data_manager.h"
 #include "components/autofill/core/browser/webdata/autofill_webdata_service.h"
 #include "components/history/core/browser/history_service.h"
 #include "components/keyed_service/core/service_access_type.h"
 #include "components/omnibox/browser/omnibox_pref_names.h"
 #include "components/password_manager/core/browser/password_store.h"
 #include "components/prefs/pref_service.h"
 #include "components/search_engines/template_url_service.h"
 #include "components/sessions/core/tab_restore_service.h"
 #include "ios/chrome/browser/application_context.h"
@@ skipping 365 matching lines @@
   IOSChromeBrowsingDataRemover::NotificationDetails details(delete_begin_,
                                                             remove_mask_);
 
   GetOnBrowsingDataRemovedCallbacks()->Notify(details);
 
   FOR_EACH_OBSERVER(Observer, observer_list_,
                     OnIOSChromeBrowsingDataRemoverDone());
 
   // History requests aren't happy if you delete yourself from the callback.
   // As such, we do a delete later.
-  base::MessageLoop::current()->DeleteSoon(FROM_HERE, this);
+  base::ThreadTaskRunnerHandle::Get()->DeleteSoon(FROM_HERE, this);
 }
 
 void IOSChromeBrowsingDataRemover::NotifyAndDeleteIfDone() {
   // TODO(brettw) http://crbug.com/305259: This should also observe session
   // clearing (what about other things such as passwords, etc.?) and wait for
   // them to complete before continuing.
 
   if (!AllDone())
     return;
 
@@ skipping 92 matching lines @@
   waiting_for_clear_autofill_origin_urls_ = false;
   NotifyAndDeleteIfDone();
 }
 
 // static
 IOSChromeBrowsingDataRemover::CallbackSubscription
 IOSChromeBrowsingDataRemover::RegisterOnBrowsingDataRemovedCallback(
     const IOSChromeBrowsingDataRemover::Callback& callback) {
   return GetOnBrowsingDataRemovedCallbacks()->Add(callback);
 }