Or perhaps the server taps a button to order food and the kitchen taps a button to clear the ticket when food goes out, and when the wait time is over a certain

Or perhaps the server taps a button to order food and the kitchen taps a button to clear the ticket when food goes out, and when the wait time is over a certain threshold, the ticket turns red on their screens and the manager notices and that is what triggers the “food is late” chain of events you experienced. (This is exactly what happens at some restaurants. No knowledge of Olive Garden in particular.)