No it doesn't. An MOT certificate is not proof of condition of vehicle at time of use only at time of test.Yes it does, it makes the car illegal to be driving on British roads... Therefore unroadworthy
A car could leave an MOT station, hit a pothole and be unroadworthy despite a 5 minute old MOT - which is why just about every motor insurance policy states that the vehicle must be in roadworthy condition at time of use and does not say that it must have an MOT.
I have only seen 2 or 3 cases in nearly 20 years where the insurance stated that the vehicle must have an MOT. In those cases the driver was driving without insurance due to an expired MOT.

). Driving a car without insurance is unroadworthy, driving a car without tax is unroadworthy and driving a car without an MOT is unroadworthy...

