Yes, its something that is in the genetics. I know Vida will trot, but she has a semi difficult time doing it and seems more comfortable in her natural gait.
I have to push her to stay in a canter as well. Many gaited people sort of frown on allowing a horse to do anything but their specific gait.
I'm assuming Harvey is similar to a Missouri fox trotter. I suggest doing some research on his specific breed and what his desired gaites are. Its best to work him in that thread of speeds rather than the non-gaited walk trot canter.
"Until one has loved an animal, part of one's soul remains unawakened..."
- Anatole France