Perhaps temperament? Saddlebreds and Friesians I have known were generally friendly, personable and intelligent. Then there is bone and feathers. Conformation I think would affect movement…particularly slope or straightness of shoulder.
I guess when you get the breeding right, you can put the right individuals together to focus on what you want as breed characteristics. In theory.
Don’t know much about Morgans or Hackneys. The Morgans I have met, while lovely, tend to be hot. But that may be my area. I understand Morgan “styles” vary in Canada by region. Have to work with them a bit to get them to settle into work…but then I may be generalizing and apologize to Morgan aficionados. They certainly fill the eye.