Accounts of ostrich cavalry from Antiquity are probably apocryphal, or they were used on a case-by-case basis like war pigs. Writers from Antiquity report that the big birds were so dumb back then, they buried their head in the sand to avoid being seen!
It's in the Middle Ages that you see documented records of ostrich cavalry. Recent studies of Ostrich DNA have found that the mutations that gave war ostriches and draft ostriches freakishly large brain sizes and leg muscles compared to dairy ostriches, first appeared around 200 CE and 600 CE respectively.
Thank the Gods for The Wolves then, they were bigger, stronger, much more obedient, and much easier to train compared to war-ostriches. They are also pretty much being favorite of children everywhere.
Yes, they did originally bred from carnivorous wild wolves, but tens of thousands of years made them adapted to basically eat whatever we have on the table.
Also unlike Ostriches, who tend to just get on business with whoever capable of riding them, wolves knew their masters, and is almost virtually theft-proof as they tend to attack strangers who tried to mount them (unless the thief your own family who stole your wolf, but then, family recognition is what made wolves being safe for even children).
Also, Ostrich cavalry are practically forced to stay mounted all the time since those birds tend to run away from the fight, while Wolf cavalrymen will dismount after the charge and their Wolves will stay and fight alongside them.