There's actually no 'producers' issued anymore, there's nothing you can produce that isn't already on recorded on a database to which all officers have access. Traffic will have remote access from within the car, beat bobbies will need to radio the details though; ANPR units will know more about you and your car than you do,
With regards to 'clamping down' (this initiative was originally brought forward to help with the illegal immigration problem you may have noticed,...allegedly) you MUST have valid insurance for a vehicle in order to give anyone else permission to drive it, and they must also have fully comprehensive insurance of their own that entitles them to drive any other car with the owners permission. There is no legal way to drive a car on British roads that isn't covered by an insurance policy in its' own right - that can be set up for a minimum period of 2 weeks if necessary. Think about it, you could buy 20 cars for example, put them in your wifes name and then get her 'permission' to drive any one of them under one policy.
With regard to driving to and from MOTs without tax and insurance: The MOT must be pre-booked and you are entitled to drive (1 ) to the testing station directly using the shortest recognised route, (2) away from the MOT station to a place of repair subject to the failure NOT making the car unsafe to drive or dangerous to other road users or pedestrians or (3) away from the testing station to where the vehicle has originally come from in point 1 subject the conditions in point 2.
You can't 'legally' drive on a public road without valid tax, or valid insurance or MOT for that matter. However, it would therefore be seriously impractical to MOT a car that you had just bought from a friend for example, that wasn't taxed or MOTd so there are clauses that allow officers to make judgement calls based on the evidence, which, if you're being honest is pretty easy to verify.