Why there is no push notification
Two remote channels were built, and both were deleted.
The first was an ntfy relay with action buttons — approve straight from the lock screen. It worked. It also handed the control plane to a third party.
The second was Bark: one line saying "there's an approval," with the control plane staying on the LAN. That was about as restrained as it gets, and it still only bought "something can page you while you're away" — in exchange for continuously telling an outside server that an operation is waiting.
Lock-screen-capable notifications must go through APNs, which necessarily means someone else's server. So the answer is a macOS local notification, fully offline, and nothing else.
The cost, stated plainly: once you walk away from the Mac, nothing alerts
you. High-risk operations wait out their timeout and get rejected. That is the
intended default — nobody should be approving rm -rf from a
coffee shop.