The text is correctly inserted. However, the problem is that you block the event loop by performing sleep/cleanup inside the closeEvent callback. Since the UI updating utilizes the event system the next paint event is not processed before closing the application.
In PyQt (don't have PySide) I was able to work around this issue with
self.processEdit.insertHtml("\n\n Stopping App.....")
event.accept()
self.repaint()
qapp.processEvents()
# clean up resources and status reports here.
time.sleep(5) # only here for the snippet
But I think this is bad practice (not only because of accessing a global variable). You should attempt to keep your GUI responsive, even during shutdown. See here.