That's a bit harsh - having no MOT (with our company) normally means we'll make some sort of deduction to reflect the reduced vehicle value if you were trying to sell it prior to the accident.
IE, you have two identical vehicles, one has an MOT, one doesn't - the one with the MOT is generally going to be more desirable.
However, is a component failure (or bald tyres) has caused the incident, and this would have been picked up via the MOT test, then the insurer does have the right to repudiate the claims.
There will most likely be a clause in your policy that states that the car must be in a roadworthy condition - however, if the car was in good mechanical condition, even if you didn't have an MOT, and the accident was simply unavoidable, then the insurer has no case to repudiate your claim.
The fact that you have a full service history and the vehicle is low mileage really helps your case.