Here is a game that is in the database where White mates on move 5.
Fool's mate is not in the database. That would be a mate by Black on move 2.
Continuing looking at the Bird opening, I see games where Black could have mated on move 4, but no game with that move in the database.
Of course, this does not mean that such mates were not found in some games; only that those games are not in the database. I think the hypothesis that very short games are not stored in the database is correct.
Also, as others have hypothesized, the cause of the missed mates that are in the database is probably because of premoving.
Fool's mate is not in the database. That would be a mate by Black on move 2.
Continuing looking at the Bird opening, I see games where Black could have mated on move 4, but no game with that move in the database.
Of course, this does not mean that such mates were not found in some games; only that those games are not in the database. I think the hypothesis that very short games are not stored in the database is correct.
Also, as others have hypothesized, the cause of the missed mates that are in the database is probably because of premoving.