The MOT is only a snapshot and therefore means that the vehicle was roadworthy at the time of the test. It does not mean that the vehicle was roadworthy at time of sale or time of collection/being driven away or in fact, even 1 second after the test was carried out.
The buyer can tax and insure any vehicle as long as it has a valid MOT. They can insure any vehicle so that would not make a difference to the case.
You rightly state that the buyer drove home after purchase. However, it is implied by the document that the buyer signed that he did so illegally and ultimately, it is the buyers responsibility to make sure the car is fit for road use before being driven.
Additionally, the seller can also state that he had the necessary repairs done on it to put it up for sale. The big red stamp is present and is unlikely to be missed. It was also signed and dated so its very very difficult to come back from that.
Having said that, no reason why you can't try going through moneyclaim.gov.uk