While reading Charles Oman's "A History of the Art of War in the
Middle Ages" it occurred to me that horses and a kind of knight were
always available to the Roman Empire,
Horses were, knights not so much, though the Romans had an "Equestrian Class", which were "knights", if not militarily, at least socially. As Tom Au says, the stirrup was the fundamental inovation that made heavy cavalry possible. Without a stirrup, riders can be unseated, not just by the efforts of enemy infantrymen, but by their own push when charging: if their weapon was a lance, the impact of it upon the enemy would backlash against them, with disasterous consequences. So, before the stirrup, cavalry was necessarily light cavalry, either armed with swords or bows. Cavalry with swords was useful for persecution, preventing the enemy from regrouping, and could also outflank the enemy, attack from behind, and disorganise their lines completely. Cavalry with bows, or mounted archers, were excellent skirmishers, that could harrass the enemy from distance, relying on speed and mobility. But neither could actually charge, in the sense we think of cavalry charges; the charging weapons in ancient times were chariots, which are far less manoeverable than mounted horses.
Medieval heavy cavalry could charge against infantry, because stirrups made unseating knights much more difficult. And its impact had high momentum: the total mass of horse and knight, multiplied by the speed of the horse. Being trampled by that was extremely lethal; only a very skilled infantry, heavily armoured, and armed with pikes could sustain a direct cavalry charge.
After this, the author proceeds to highlight that cavalry was a major
advance in the art of war after the Roman Empire, and that mounted
knights always won battles against infantry. This created my doubt: if
horsemen were so superior in battle why were the legions composed
mostly of infantry?
Oman is basically right about the power of knights; of course they didn't win "always"; Agincourt is an example of them being soundly defeated. But Agincourt is one exception; several different and unusual factors made the English victory there possible: the narrowness of the terrain, the muddy soil, poor command, the disorganised structure of the French force. Those, combined with correct tactics by the English (the ample use of longbows by bowmen protected by spikes), defeated the French cavalry). Also, heavy cavalry is not a good option in hilly terrain - which is the reason that plebeian Switzerland could secure its independence against foreign nobility. But, "normally", cavalry could break the lines of infantry, yes.
Tom Au brings up the interesting notion of rider and horse becoming a single unit, due to the stirrup. Similarly, infantry was not an actual unit at that time. Bowmen and pikemen were different weapons at war; the former could shoot, but only the later could engage directly with the enemy. And that intrinsic division was to be only healed much later, with the invention of the bayonet, that allowed footsoldiers to charge and shoot simultaneously (and, in this respect, Agincourt is also kind of a premonition: due to the disarray within the French army, bowmen could, at the end of the battle, take swords and finish the defeated Frenchmen; but such fusion between the two parts of infantry was, at that time, only possible in extraordinary circumstances).
Something that Oman says is that after the Roman Empire there were
mounted archers instead of mounted infantry. Was that the 'major
advancement' in military technique?
Mounted archers are much older; the Romans faced them when they fought the Partians. And Huns and other nomadic people also used this kind of military unit. The specificity of mediaeval cavalry was the use of brute force and lances, to turn knight-and-horse into an almost unstoppable impact weapon, that sought direct combact instead of diverting from it.