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.