My experience has been like Joed's.
My wife rides a quarter horse, and it's walk, trot, walk, canter, etc.
My SSH in the flat walk, forces the quarter horse to trot and in a running walk, he has to canter.
My SSH doesn't lolly gag around but rather walks with a purpose and most non gaited horses struggle to keep up.
You might as well "Ride With Pride" on a Tennessee Walking Horse or Spotted Saddle Horse.
For we wrestle not against flesh and blood, but against principalities, against powers, against the rulers of the darkness of this world, against spiritual wickedness in high places.