Revert of Delete "shutdown without closing browsers" path

chrome/browser/lifetime/application_lifetime.cc

Index: chrome/browser/lifetime/application_lifetime.cc
diff --git a/chrome/browser/lifetime/application_lifetime.cc b/chrome/browser/lifetime/application_lifetime.cc
index ca2b639c2160140a8c0aa8810645e74cb36f6600..b7eb115271084b7b239b1f0462691a1567a25e9e 100644
--- a/chrome/browser/lifetime/application_lifetime.cc
+++ b/chrome/browser/lifetime/application_lifetime.cc
@@ -113,8 +113,9 @@
   // If there are no browsers and closing the last browser would quit the
   // application, send the APP_TERMINATING action here. Otherwise, it will be
   // sent by RemoveBrowser() when the last browser has closed.
-  if (chrome::GetTotalBrowserCount() == 0 &&
-      (browser_shutdown::IsTryingToQuit() || !chrome::WillKeepAlive())) {
+  if (browser_shutdown::ShuttingDownWithoutClosingBrowsers() ||
+      (chrome::GetTotalBrowserCount() == 0 &&
+       (browser_shutdown::IsTryingToQuit() || !chrome::WillKeepAlive()))) {
     // Tell everyone that we are shutting down.
     browser_shutdown::SetTryingToQuit(true);