So do their riders. But it does not change the fact that the horse is the real athlete.
The only sports I can think of where a woman not only competes on an equal footing with men, but may actually excell, are shooting, archery, darts, pool … and maybe horseshoes. Sports that require finesse and concentration, not strength.
Yes, in mixed doubles, not head to head. Undoubtedly a few women could beat most men one on one, but still it is extremely unlikely that a woman could win the tournament no matter how enlightened you think an open gender tournament might be.