I understand where you are coming from. I agree that a breeder should breed horses that are easily handled and have good natured as stallions. That's a nature/nurture debate. A good breeder starts out with a well mannered stallion that produces well mannered git. Is that the result of genetics or good handling?
What I mean by personality is that a stallion is not worth much if you can't handle him enough to show him, if you can't safely have him around other horses. If all your stallion can do safely is stand out in a field and naturally cover mares there is no real value in that stallion. If your stallion has talent, great conformation and awesome blood lines but is a hazard to have at a show than the other three go to waste.
You are correct we do see horses change personality wise with training. That said, its a hard row to hoe and testosterone can add to that complication. Some horses are more willing than others. To have the work be worth the pay off, the stallion would in my opinion, have to be a one in a million horse being owned/trained by that very rare combination of experience, perseverance and skill. A dangerous horse is not worth having, stallions are no exception to that. I have seen a lot of bad behavior tolerated by stallions because they are stallions (biting, head shy, not leading well, etc).
Performance is great but personality and willingness are the cornerstones of performance.