I have no answer for the exact number of horses that died in the service.
On the other hand, it’s just as likely that no horses died from exertion. You have an incorrect idea about horses, probably based on what you’ve see in the movies and TV about horses in the military and the old West.
Of course, horses were regarded without any of the concerns that we might have now. They were domestic animals like cows. You could raise them for meat, but they were more valuable as transportation. Some people recognized that mistreating a horse was a poor use of assets, while others didn’t care. There was less romance about horses than today.
Still, knowledgeable riders in the West and in the military realized that horses were a precious commodity. A horse can gallop for a length of time measured in minutes. It can move faster than man can walk for hours at a time, day after day, carrying substantially more weight than a man. It is this stamina, rather than speed, that makes a horse valuable.
Even so, a knowledgeable rider would get off and walk his horse from time to time to let it rest. In a distance of 10 miles, this wouldn’t be necessary, but cavalry columns did it several times a day when moving cross-country. A John Ford movie (I think it’s “She Wore a Yellow Ribbon”) shows soldiers doing this.
Cowboys had to be on their horses most of the day, so walking them wasn’t always an option. Instead, they used “remudas” or packs of horses. As needed, they’d pick out horses from the ranch’s stock and add them to their personal pack. Then they’d switch off on a regular basis to a fresh horse, letting the “worn” one out to graze and rest. The horses were herded and handled right along with the cows.
I would guess that Pony Express horses were ridden at a fast trot between stations. That’s a day’s good exercise for a horse, but no more. A horse wouldn’t be ridden every day, either.
Stagecoach horses were handled the same way.
In the city, some stupid tradesman might abuse a horse in anger, but in rural areas a man wouldn’t dare. If the horse collapsed, the rider would be stuck miles from nowhere with a lot of walking to do.