I’d still go racing for the best quality stuff. But every country would need to be the same, because where gambling existed there’d be more prize money, attracting the best horses…
But let’s be honest here, even if there wasn’t any official gambling the illegals would soon move in. With non-regulated betting, no paper/computer trails, bets made with a shake of the hand… you’d get far more horses being "stopped".
But in the spirit of the original poster…
If there was some way of magically ending man’s wish to gamble… Sure it would exist, but in a massively reduced state and lack of atmosphere. Reduced prize money and no gambling means there is no other way for connections to make money. Therefore, fewer owners, trainers, jockeys able to make it pay, so fewer horses. Which as Paul says wouldn’t be a bad thing. Though it would be a fraction of what we have now, probably too small.
Back to a pastime for the extremely rich. The owner-breeder might even make a come-back.