It's not just a weight thing, although of course that would be huge, considering jockeys typically weigh around 100 pounds! It's simply the rules of the game. Gotta have a rider aboard, at least as the horse crosses the finish line, which then raises another good question.
What if, like we may see in an old comedy film, sitcom, or old-school cartoon, a rider falls off a horse, then somehow manages to get back on and guides the horse to victory, or any any part of the purse (2nd, 3rd, 4th), for that matter? Would the result stand? I'm guessing yes, it would.
I believe the rules, variant as they are, would require the rider AT ALL TIMES.
Again, the bottom line is weight. Losing 100 lbs even for a brief part is a huge advantage.