It's a knotty problem! I think, at the heart of it, there's more than one piece of legislation at work simultaneously.
Most MOT fails are contraventions of the Construction & Use Regs and the Road Vehicle Lighting Regs. So it is illegal to use a car, (whether it has an MOT or not) with a defect that falls foul of either of those regs.
Completely separately, it is illegal to use a car without a valid MOT (assuming it's not under 3 or over 40 years old, of course).
What I think happens, is this:
(a) if you fail an MOT for something that isn't "dangerous" before the expiry date of the old one, you still have a "valid MOT certificate" until the old one expires. You can still drive the car until the expiry of the old one.
(b) if you fail the MOT for something that is "dangerous" before the expiry date of the old one, you also still have a "valid MOT certificate" until the old one expires. However, if you drive the car until the expiry of the old one, you would be committing a separate offence under the C&U or Lighting regs (and maybe something under the Road Traffic Act too), but NOT under the MOT regulations, as the vehicle still holds a valid MOT certificate. This is bourne out by the fact that when you go online after failing for something "dangerous", your MOT still has the original expiry date.
I therefore don't believe that the MOT tester has any legal power to prevent you from driving the vehicle away, but if he was feeling vindictive, he could call the local "Plod" and tell him that vehicle, registration number ABC 123, has just left his station with (say) no brakes. If Plod could be bothered, when he caught you, he could do you for the breach of the C&U regs, but not for failing to have a valid MOT.
Note that the old "GV9" prohibition notice for a dangerous defect, is peculiar to commercial vehicles. There is no equivalent for cars.