As a practical matter, many do. If you hunt, for example, you've probably field dressed a deer, elk, pheasant, or feral hog. Some farmers still slaughter their own meat, although most will pay a locker to do it for them. I've witnessed the industrial slaughter of cattle, hogs, and buffalo, as well as the butchering process in country lockers. I've also seen the parts that get sent on to become dog food.
What bothers me are bad slaughter facilities (like Agriprocessors, a few years back). Those who can't /won't find a rescue, or who simply want some value returned on an unwanted/unneeded animal currently subject the horses to long rides to slaughterhouses in Mexico or Canada. Standards and oversight are simply lower. The fact that it is so difficult to dispose of a horse has accelerated the replacement of horses with 4-wheelers. As demand drops, more horses can't be sold, and aren't needed, and the cycle repeats itself. It'd be nice if every horse could be treated like a beloved family pet, or sent to a rescue. That's simply not reality. If New Mexico won't allow horse slaughter, then owners will continue to ship outside the country, or they'll look to open a facility in a Native Reservation as a means to circumvent the law. It's like drugs: I don't like 'em, but I'd rather see them legal and controlled.