Earlier quoted context omitted.
But what’s the point of catching an overstay if they’re leaving already? It’s more important to catch them when they try to re-enter (at which point you could have collected data from airlines to estimate whether they previously overstayed). Exit checks are pretty pointless if there’s no violation that would lead to enforcement other than deportation (since the traveler is already self-deporting).
If someone has overstayed you might want to note that down in case they come for another stay, right?
if you don't get any record of them leaving, from a manned border crossing or an airline reporting to you, then they've overstayed