I would look for a QH or Appendix. You will WANT scope, considering what you are looking to do, and the price should be better with a QH bc they are simply so many of them. I also, IMHO find them calmer than anything more blooded.
SO MANY of us trail ride our horses Western Pleasure, but SB, though great horses, are breed for gait and style, not multi-disciplines. ANY sound horse can pop over a 3' vertical, but a gaited horse isn't built to couple and "take you with him". I currently own two KMH's, but I don't jump them. I will be training my QH to jump this year, and we'll see what he has in him.
BTW, many splashy horses are bred just for their color, not for their build.
Plus, if you show, you will be riding among a sea of paints. You would really stand out if you bought a solid brown/black horse with no blaze and no sock/stockings at all!
It sounds as though you think Saddlebreds are gaited. They are NOT. They are trained to gait.
One I owned, was a five gaited champion in his day. I rode him doing everything but gaited.