I see the logic behind cloning a stud or broodmare.
The clone will not have the training, fitness and lifestyle of the champion. Therefore it will not perform as the champion would.
But what is the point of being a stud? To pass on DNA. The training, fitness and lifestyle of the champion cannot be passed on even from the original horse. Only the DNA can. The DNA in the clone is exactly the same; hence the clone provides the same benefit to its offspring as the original horse.
Of course, this is not strictly true, as clones can have problems with the stability of their DNA, but broadly it is true. Cloning a horse is an extremely difficult and expensive process, but with a high-value stud it may be economically viable.
In terms of naming and registering, I do not believe the horse should be registered as its source, but as a different horse. Nevertheless, the logic behind breeding to it is the same as the logic behind breeding to the original champion.
If you believe everything you read, better not read.