The US horse population has varied somewhat over the years, but there are currently 6.9m. That is far from a dwindling to nothing. To give a sense of proportion, in 1867 there were 8m equines, looks like a peak of 21m in 1915. There was never any widespread killing of horses, and most horses are not allowed to breed anyway, whether populations are rising or falling. You can find some excellent references here
There is a thriving wild horse population, and the domesticated ones are split between work, show, and leisure animals. The vast majority of them have happy lives. There are still literally millions of horses doing farm work, a hundred years after tractors became common.
I'm pretty sure your dystopian view of the future is also wrong. I think humanity will always need to guard against abuse of power, but the trend over our history is towards more freedom and a better life.
People forget how much things have changed. A hundred years ago nearly everyone worked on a farm. Now all those people lost their jobs but somehow found something else to do.