Being registered means diddly squat when it comes to diseases, joint problems or health. There is no scientific proof that a registered horse has less health issues than an unpapered one, IT'S A NON ISSUE!
I raised beagles for over 25 years. Drove all over the country to breed to the stud I though would produce on my bitch line. Here are a few of my observations.
I've taken very good papered bitches and bread them to some of the best studs in the country. They have 3 pups, and given the very best of care 2 die. A grade bitch, wormy as all get out, and never seen a vet, will have 13 under the front porch steps, raise them in the dirt and all 13 will live....
Breed the best to the best and there's no guarantee the offspring will be of the same quality. I've probably bread 150 or more beagles, had 3 truly outstanding ones, about twenty pretty good ones, and the rest were culls and couldn't run a rabbit 100 yards.
That said, after it's all said and done, I think I had more fun with the 25 dollar dogs I started with.
No doubt to reach the pinnacle in competition registered stock will get you there quicker because it's carefully bread for a purpose.
If you want to go to the Olympics then a grade horse isn't going to get you there. Same goes for the Kentucky Derby. If that's what you want the the answer is no. Can you have fun and show in most local shows? You bettcha....but as the level of competition rises you'll most likely be excluded due to lack of papers.
Many grade horses are out of registered stock to begin with. You can't ride papers.