I don't understand the logic behind this. Why would people not care what the stallion can actually DO? Can you elaborate more on why this is so?
While many stallions do compete really well under saddle in mixed company there are plenty of good stallions that don't and get too easily distracted by having mares around them. You will find that many stallions in the UK are never competed other than 'in hand' yet they still produce top riding stock and many of the HIS (Hunter Improvement Society - now the Sport Horse Breeding Society) stallions were TB's that had only ever raced. They were/are selected for conformation and paces (how they move) and their ability to deal with situations - eg you look for a horse that's bold and outgoing if you want something to compete.
The UK show ponies all trace back to Arabians and TB's selected for looks, temperament and movement.
A lot of owners will be breeding from a mare that's already proven itself in some way and retired to breeding.
I used several TB racing sires on my mares because their progeny was proof that they were throwing good stock for the competition ring. If a stallion has excellent conformation and the paces to do the job and the right attitude then how well it turns out is really up to the person that trains and rides it.
It doesn't matter how well a stallion has performed under saddle if you use a poor mare you are not going to get a winner and even if you used the best mare in the world there's no guarantee the offspring will perform well if its not trained & ridden correctly or lacks the personality and drive
You'll see much the same in breeding racehorses, I had a son of Nijinsky that was out of very good mare, he was trained on a top yard but could barely be bothered to leave the starting gate in a race.