Why don't my pushed tasks re-sync or update in the task app after I push them?

That's by design — it's the hand-off model. Once handed off, your task app owns it. (An earlier "live mirror" version that tried to keep both sides matched caused jumpy behaviour and duplicate tasks, so it was removed.) If you want a fresh copy in your task app, drag the item out of the "Pushed tasks" folder on your device to reset it, then push it again — that creates a brand-new task.