Why does Cogburn kill the horse?

The following is from the script at this link.

It would seem that they had ridden Little Blackie beyond a point of restoration, even choosing to cut the horse with a knife to drive it on. Given the foaming, and the unsettling noises coming from within the horse, Rooster chose to end the horse's misery. The horse was nowhere near a place it could heal. In Rooster's mind, he most likely thought it wouldn't even if they were close.

THE MEADOW

Mattie is woozy. As Little Blackie crosses the field at full gallop Mattie looks blearily at the littering bodies of horses and men. Next to Lucky Ned’s body his horse, saddled and riderless, swings its head to watch as Rooster and Mattie pass. Mattie’s eyes are closing.

LATE DAY

Mattie’s eyes half-open. Little Blackie plunges on, through a rough road in woods, but slower now, his mouth foaming. Rooster "Come on, you!" Mattie "We must stop." Little Blackie is played out. Horrible noises are indeed coming from the horse, but Rooster is grim: Rooster "We have miles yet." He leaves off whipping the horse and takes out his knife. He leans back and slashes at the horse’s whithers. Little Blackie surges. Mattie screams. Mattie "No!" A locked-down shot as horse and riders enter at a gallop and recede.

NIGHT 113

It has started to snow. Mattie is flushed and soaked with sweat. The horse is laboring for breath. Rooster gives inarticulate curses as he kicks it on. Mattie looks ahead: Barely visible in the moonlight a man mounted bareback rides on ahead. A sash cord holds a rifle to his back. He recedes, outpacing us, disappearing into the darkness and the falling snow. Mattie "He is getting away." Rooster "Who is getting away?" Mattie "Chaney." Rooster "Hold on, sis." Mattie is falling. It is unclear why. Her legs squeeze the horses flanks. Her hand tightens on the horses mane. Rooster’s arm reaches around to hold her. Little Blackie is giving out, going to his knees and then all the way down. Rooster hangs on to Mattie as the horse sinks. He pulls her clear, lays her on the ground, and then steps away from her, taking out a gun. The horrible noises coming from the horse end with a gunshot. Rooster reenters to pick up Mattie but she screams at him and claws at his face, opening fresh gashes. He ducks his head as best he can to avoid the claws but that is the extent of his reaction.

Ironically, it is an act of mercy. Cogburn knows that the horse is doomed to die. The horse will either starve to death, or get torn apart by predators and eaten. Cogburn shoots the horse to put it out of its misery, and save it from going through the torture of a slow, horrible death.

A horse with rider can be panicked into riding itself to death due to exhaustion either through the inability to breathe or cardiac arrest. Horses are prey animals and their natural instinct is to flee and a rider can keep the the horse in a somewhat panicked state. There is a joke that describes this: A horse must be coaxed to stop going. A donkey must be coaxed to start going.

The script indicates Little Blackie is making terrible wheezing noises which implies Little Blackie may be having breathing problems and may be exhausted almost to death. Thematically, Little Blackie is shown to have True Grit and pays a price for it. (Or pays for Mattie's grit depending on how you look at it.)

The other plot question is why Rooster didn't take Ned Pepper's now riderless horse and bring it along. The extra horse, without the weight of two riders could, theoretically, have been swapped out for Little Blackie half way through the ride, extending their range.

Also, Hattie indicates Little Blackie is a pony. Ponies are horses bred for a certain temperament and build. They tend to be smaller but have more endurance for their size, so they tend to be work horses more than ridden horses (this is why the auctioneer at the beginning of the film wants the grey horse but not the ponies. It seems work horses are harder to resell than riding horses.) Because ponies have a smaller size, Ned's larger horse may have been a better choice given Rooster's larger size plus the weight of Mattie. However, Little Blackie is a "character" in the plot so her death is more emotional and Rooster is in a state of panic and may not have been thinking.