There are two entirely separate pieces of legislation. One says that it's illegal to drive without a valid MoT certificate; the other says that it's illegal to drive an unroadworthy vehicle.
Your car could fail a test taken a month early with a shattered windscreen, four bald tyres and not a single light working and you could still drive it without contravening the first piece of legislation (because the 'old' MoT certificate remains valid). It's only the second piece of legislation which would be contravened.
So it's not enough for the police to know that a vehicle has failed its MoT. They would need to show that the vehicle was also 'unroadworthy'. Some things which would result in an MoT failure (such as a cracked windscreen) don't necessarily render a vehicle 'unroadworthy', so it's completely legal to drive such a vehicle while the old certificate remains valid.
Chris