Warhorses Are Specialists

The old saying was "Stallions to the battlefield, mares to the plow, and geldings to the packtrain." (Hyland, Equus: The Horse and Mule in the Roman World)

Medieval horses were classified in armies into dextriarii (destriers, high-class warhorses), equii (ordinary medium warhorses) & rouncii (rounceys, light warhorses). All of these were stallions until the Renaissance when they started using mares that had been proven sterile. This is important to note: mares would cause stallion fights, otherwise, whenever they came in heat. So the warhorses of the Conquistadors could not have bred, if lost or abandoned. The Europeans had to bring in breeding mares to raise horses in the New World. They kept them away from armies.

Geldings are almost always too subordinate for ancient or medieval cavalry. On the other hoof, the modern cavalry tended to be mostly geldings with some mares because they wanted less aggressive remounts.

You can throw a bundle on your destrier and lead him, but horses must be trained to all their unnatural work for humans.

Horses to be driven, to be used in harness for draft, must be taught to draw before they are taught to be ridden. Your destrier will not know how to draw, so you can't use him to pull a wagon or artillery (MacDonald, Driving). You can't suddenly teach him or bully him into it. The resistance of the collar, to him, is a clear signal to stop. He just stands there getting more confused & irritable. An irritable destrier can kill you where he stands.

On the other hand, your farm horse can carry you somewhere, but he lacks all the warhorse training. He will spook & bolt at an unfamiliar kind of animal - say, if he hasn't met sheep or llamas before, let alone camels or elephants. He will be terrified of loud sounds - shouting, gunfire, trumpets. Bright or glittery things will send an unaccustomed horse right off. The smell of blood makes him leave the area - he's a prey animal. You have to train packhorses to tolerate bloody loads to take them on hunting trips.

Also, the farm horse will not know the physical signals from horse to rider. Warhorses must neckrein, besides listening to leg pressure and weight shifts. The rider's hands are mostly tied up with weapon and shield. There is a very coercive bit, rarely used, to control a horse gone over the edge. (Hyland, The Warhorse) American Western riding, from spade bits to long stirrup leathers, is a repository of Ren combat riding, inherited from the Conquistadors.

The destrier has even more training. As far back as the Persian Wars, the horse of Mardonius would fight "on its own" and S. A. Bolich has tracked purpose, in combat, for all the "airs above the ground" surviving in the Spanish Riding School of Vienna. At the very least, the destrier had the training that got him his name: going on a dexter lead (our left lead) so his balance was good when lance met shield.

Now, in the Modern cavalry, say, Napoleonic, the officers would provide their own mounts. One of these might be a ride-and-drive who had been taught to pull a cart before being backed (trained to saddle). It would be rare. (Badminton Library of Sports, Driving and Riding) Officers generally wanted a Thoroughbred hunter under them, for clearing obstacles.

This isn't even going much out of Europe. My information is thin on, say, if Sarmatian horses were mostly ride-and-drive or not, and while I've read The Horse in West African History, I have no clue to their army arrangements.