Everything has a genetic component to it.
Your best prospects are going to be bred to do the job or event you want them to do. Speed, agility and a good work ethic are ALL very genetic.
They are going to have the conformation to do the job you want them to do.
They are still going to have to have proper training. The better the training and conditioning they get the better they will do.
They are still going to have to have 'heart' and try, a good work ethic and a willing disposition -- also all with a big genetic component.
Can horses that are not bred to do something still be good at it. Sure. Once in a while you find one. But the odds are waaaay against finding one that can beat the horses that have all of the above. The more breeding is specialized toward a specific event, the better these horses have gotten. Cutting horses that won the World a couple of decades back would not even make the finals in a big event now. Same is true for barrel racing, reining, jumping, etc. The level of competition nowadays is light years above what it was when I started training and showing.
Look at it this way. There is probably a Beagle somewhere that would learn to work and herd sheep or cattle. But do you think it would easy to find one that could beat a Border Collie? Not real likely and you could go through 100 looking for one that would herd at all. It would probably be about as easy to get a bird dog to tree a coon.
To get a horse with the athleticism and pure speed to run the barrels at a good level (something above 3D) you need a horse with natural speed, agility and the desire to run and use that speed. It is just a lot easier and a lot more successful results come from a horse that is bred for the job.
In the 50+ years that I have trained horses, the biggest single improvement I have seen in performance has come from the breeding of better prospects. Right behind that comes the better training techniques that have come from young trainers apprenticing to World Champion trainers instead of trying to invert it over and figure it all out.