They're happy because they are partaking in a hobby with an animal they love deeply and have invested a lot of time, money and care into. You don't ride horses unless you love horse riding, that's all that is making them happy.
No you don't cum in the saddle just because your genitals are in contact with it. For one, that 'rocking backwards and forwards' isn't pleasureable the way grinding on something would be. It's not the right motion, for lack of a better term. For another, you're far too focused on all the little things that use up your concentration while maintaining control of an animal underneath you to orgasm. Even if something was getting you turned on (which is doubtful as it's not a 'nice' kind of contact), you'd soon be distracted. You're not just sitting there, you're working constantly to control and direct the horse (not to mention to keep yourself in the saddle).
When your horse is walking, your hips are moving with it and your crotch is stationary in the saddle. When trotting, your crotch is getting smacked in and out of the saddle as you stand up and down in the stirrups. When cantering, again your hips are moving with the animal and your crotch is in firm, stationary contact with the saddle (not rocking back and forth or grinding against it). When galloping, your crotch is not in the saddle at all. So really, your questions boils down to would a women be able to orgasm while sitting still on a hard leather seat or while whacking their clit against it, in both cases while their mind in completely focused on a task that requires them to remain attentive and in control or risk injury to themselves. No. Not really.
That's my experience anyway. No stimulation other than having your crotch pressed against a saddle and having your mind occupied isn't the key to orgasming.