I know a few people who own TWHs but never gait them - just dog-walk the whole time on trail.
Some of them have never owned gaited horses before, they don't know what the gaits are supposed to feel like, so any time they do more than a dog-walk and the horse is pacey, they just let the horse stay pacey.
They ride the horse so balled-up into the bit that the horse can't even head-nod, there is no freedom of movement in the shoulders, and they have the horse's hooves trimmed so upright (like a stock horse) that it just moves different.
After years of doing this, the horses just pace all the time. It's a shame. . .but they're what I'd consider a non-gaited TWH. They might as well just buy a non-gaited horse if that's how they want to ride.