Don't confuse the requirement for correct history for a classic road car and the specifications needed for a pure competition car.
For a 'genuine' Cooper 'S' to have a high value and be considered as 'original' it should have the original V5 together with matching numbers of VIN and engine together with the correct age-related specification for all major components.
A true competition car will have a log book issued by the MSA or equivalent body, have some competition history and meet the requirements for the category of motor-sport into which it will be entered.
For most true competitors a Mini will be considered as a 'Cooper S' so long as it has a Mk.1 or Mk.2 bodyshell, twin-bolt front sub-frame, a proper 'S' block with a remote gear shift, 'S' brakes with 10" wheels, etc. It may well have been built up from parts as most race cars were in the 1960's. After all, why take a complete Cooper 'S' and strip everything out when one could buy, say, a damaged Mk.1 850 which has an identical basic bodyshell and sub-frames, and from that buy the specialised race parts needed and build a car which will both pass scrutineering and be competitive. Virtually all the Mini race teams built their cars from scratch using parts from BMCSpecial Tuning or Comps Dep't. This also applied to rally cars and rallycross cars as well as pure race cars.
The car for sale here is most definitely a car which meets or met the race regulations of the day and it looks to be good value. Obviously one can't take such a car and put it on the road with a real 'S' V5 and it it was to be road registered it would likely get a 'Q' plate.
I hope this helps.