Sounds like the OP had the same issue I did with Selena.
I'll give you my little personal experience.
Selena was also a reiner. She was finished when I got her and won money being shown as a three year old, and I got her when she was four. She was lazy. I mean, kicking, whipping, spurring, and you got to trot. Without spurs or split reins in hand? No way in hell you were getting her to even lope. She was a safe, consistent reiner but she wasn't a lively, run-jump-slide sparky reiner. She was correct but not exciting to watch or ride.
If you would of told me two years ago she would be my main barrel horse and be winning, I would of laughed. No way she had that type of energy.
Well, I started working on her in lessons with other people. We worked a lot on speed transitions, so when we were asked to speed up and the other horses would pass her..She got mad. She wanted to be in front. So she sped up until she was in front. She stretched out and just flew. Ever since then, her speed has improved. But he had to learn how.
Still, at this speed, she wasn't as fast as she is now. Faster, yes, but not fast. She had to learn, especially in confinement, that she COULD run and really stretch out without hitting a fence.
Here's what I did.
Once or twice a week I would breeze her. That means, walk, trot, lope, then push for the speed and make them run. Then back to the lope, trot, walk. She had to build those muscles slowly and learn to run. She got into running shape quickly with this added onto her. She sleaked out and learned what she could do.
She still shut down between barrels. What I would do then is push her hard to the barrel and not turn, just run past it and run her along the rail. She learned that sometimes she wasn't going to be asked to turn, so she started stretching out before. Then I started adding more of the turn. She learned how to do these things and made an exceptional transition from a reiner to a nice little 3/4D horse (And getting faster)