I'm also about 6'3 and 250, I have a 15.2, 1100lbs, Paint who does fine with me and also a 15.3 Quarter Horse gelding at about 1250lbs. He'll handle me fine but he's a young horse I don't trust as much as the paint.
I would stay away from the Draft breeds , they are gentle but your (my) hips get sore because of such a wide girth.
I love the Quarter Horse breed (a Paint for this purpose is QH with too much white, same breeding, type, and temperment. So for the sake of argument for now consider a Paint to be a QH. But a Pinto is not always a paint, some pintos can be very hot horses, i.e. Saddlebreds, National Show horses) many very sound, sane and safe QH's out there. But lots to know before you make your decision, for example you can find a Reg. Quarter Horse that is mostly Thoroughbred, and could turn out to be a very hot horse, hard to hande and dangerous for a beginning rider.
Do lots of research, Arabs a smaller horses that are hot, Thoroughbreds are big and hot, As I stated QH and Paints allow Thoroughbred into the mix. But, lots of very nice mostly "foundation" QH's out there. I too love mules, but they are very smart, can be stubborn and can kick you into the middle of next week. Appy's can be good but most I've seen can be hard headed and stuborn (not to offend appy people). Not much experience with walkers and Paso.
Good luck, do lots of reasearch and remember any horse can kill you any day. Learn how to handle them how to be safe and keep yourself a way to get out of any spot (always be between the horse and the door if in the stall). Learn how to read a horse, what ear position means what it's posture means and actions with it's mouth means.
Good luck and welcome to a wonder experience, Horses, with dogs are here to be a companion to men and women.
Alan