Were mounted soldiers really an advancement during the Middle Ages?

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 hors*m*n 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.

Mounted soldiers date back at least to the Neo-Assyrian Empire, and were a significant part of Persian armies, who emphasized heavily armored hors*m*n (including armor for the horse); the general term that gets used for these pre-medieval heavy cavalry is cataphract.

It's true that the Romans didn't emphasize cavalry much, but they did end up deploying some heavy cavalry units in the east, against the Parthian and (later) Sassanid Persians; the Byzantines continued this practice in their wars against the Sassanids.

The idea of stirrups (which showed up long after armored, lance-using cavalry appeared in the Levant) as a critical invention is generally now seen as an exaggeration. (Stirrups are better, certainly, but mounted warriors were able to brace themselves and deliver powerful attacks using well-designed saddles prior to the appearance of stirrups.) See here for a list of references on the stirrups issue.

As for mounted archery: it certainly was a significant advance -- groups like the Mongols and Turks used it very effectively, and tended to defeat European knights when they fought against them -- but again it's something pre-dating the Romans, as Luis Henrique noted in his answer. There are clear records of it from the Neo-Assyrian Empire.

If you want a nice illustration of when these things were actually in use, there's the Battle of Carrhae (53 BC), where a Roman army was wiped out by the Parthians using a combination of cataphracts and horse archers. (But note that the Romans also won battles against the Parthians and their successors the Sassanids, so the "cavalry always defeats infantry" argument is clearly false.)

Soldiers mounted on horses were an advantage when horse and rider could fight as a unit. This occurred after the invention of the stirrup, up to the time of rifles.

Before the stirrup, a cavalryman had to manage his horse and weapon separately. That gave the advantage to "infantry," at least the Roman kind. The stirrup was not introduced into European warfare until at least the fifth century CE (at the very end of the Roman empire), possibly later. Basically, the Roman empire did not live to see the benefits.

The addition of a stirrup basically fused horse and rider. A horseman could charge infantry at 4-5 times the speed of the footmen, and the horse also weighed 5-10 times as much as a man. It's no exaggeration to say that on level ground in good weather, a horse and rider working as a "team" had 5-10 times the value of an infantryman, when both were armed with spears, lances, and swords. This gave cavalry the advantage from 400 or 500 CE to at least 1500 CE.

The advent of the musket reduced the horeseman's advantage (in much the same was as the long bow), but it wasn't until the introduction of long-ranged rifles in the 19th century that the advantage of the horseman was fully overcome. It was much harder for a horseman to manage a rifle and reins simultaneously, than a blade weapon and reins, so horse and rider effectively became "two" (different) units again during the rifle era.

On the other hand, hors*m*n could use blade weapons during the era of the musket (1500-1800), for which "killing distance" was only 50 yards, at which range the musketeer could fire only one shot. It was the longer-ranged, faster-loading rifle that made it impossible for a horseman to compete on equal terms with infantry.