Fix flakiness in SystemDisplayApiTest.SetDisplay

RulesRegistryService posts tasks to the IO thread which access
ExtensionWebRequestEventRouter::GetInstance().
ExtensionWebRequestEventRouter accesses ExtensionsAPIClient::Get(),
which used to be destroyed before the tasks could run in some
situations. This change moves destruction of ExtensionsAPIClient
to ShellBrowserMainParts::PostDestroyThreads(), which should be
safe since tasks will no longer run at that time.

Bug: 866904
Change-Id: I974d474d08ff9d7c0c727a7c95d26f04363ce7f0
Reviewed-on: https://chromium-review.googlesource.com/1194579
Commit-Queue: Clark DuVall <cduvall@chromium.org>
Reviewed-by: Karan Bhatia <karandeepb@chromium.org>
Cr-Commit-Position: refs/heads/master@{#587343}

extensions/shell/browser/shell_browser_main_parts.cc

@@ -279,8 +279,6 @@ void ShellBrowserMainParts::PostMainMessageLoopRun() {
   BrowserContextDependencyManager::GetInstance()->DestroyBrowserContextServices(
       browser_context_.get());
   extension_system_ = NULL;
-  ExtensionsBrowserClient::Set(NULL);
-  extensions_browser_client_.reset();
 
   desktop_controller_.reset();
 
@@ -302,6 +300,9 @@ void ShellBrowserMainParts::PostMainMessageLoopRun() {
 }
 
 void ShellBrowserMainParts::PostDestroyThreads() {
+  extensions_browser_client_.reset();
+  ExtensionsBrowserClient::Set(nullptr);
+
 #if defined(OS_CHROMEOS)
   network_controller_.reset();
   chromeos::NetworkHandler::Shutdown();