So actually, it had. It's a popular misconception that couched lancing techniques can't be performed without stirrups, but they were. We know this because we find couched lancing on art going back as far as the Achaemenid Persian era, and it can be seen here on the first century B.C. Orlat Battle Plaque. What enables couched lancing is the high-backed steppe saddle, or the Roman four-horned saddle, not the stirrup (which just assists in mounted archery more than anything else).

However, late Roman and "Byzantine" cavalry warfare was fundamentally different than medieval cavalry warfare. This difference was because the Romans adapted their cavalry from the Alans, Huns, and Avars, and trained them to fight in the style of the steppe nomad. Fundamentally, this means that the Romans cavalry almost always doubled as horse-archers, while western cavalry did not. This was also the fundamental difference between western shock lancers, and Roman kataphraktoi or klivanophoroi.

The 6th century Strategikon gives us an overview of the Roman art of "lance-and-bow" warfare. The Romans divided up their cavalry into two groups: the koursorses and defensorses, or the attackers and defenders. The koursorses would attack first with their composite bows, riding out towards the enemy and firing, then wheeling around and performing the "Parthian shot" in the "Alan drill" as the Romans called it, attempting to lure the enemy out of their protective formation and disorganize them. Part of the way through the drill, the Defensorses would begin to advance, so that when the Koursorses split and wheeled around in either direction, they would come through the gap to attack with lances in traditional shock tactics. The Koursorses would switch from the bow to the lance, and then come around on the flanks to hit the enemy. Meanwhile, a third and fourth group of cavalry would perform different tasks: the flank guards and outflankers. While the flank guards stayed back to protect the flank, the outflankers would attack the flank of their opponent while the Koursorses and Defensorses engaged. Thus the Romans fought with a three-pronged attack strategy that when properly executed would devastate their opponent.

However, this is not to say that every Roman cavalry unit operated this way. During the time of Leo's Taktika in the 10th century, the Roman kataphraktoi and klivanophoroi became more specialized towards the shock lancing role. And the Romans maintained units of light screening cavalry armed with javelins, or dedicated horse-archers. Usually these troops didn't come armed with the long cavalry kontos but a shorter variety. They were still capable of being lancers, they just were specialized for a very different role: scouting, harassing, staying out of the melee and only entering it to chase down the routing enemy. During the late Roman period we also see cavalry armed with the sword and a larger shield, scutarii the Romans called them, whose purpose isn't exactly known but they may have been something akin to "Dragoons" in their role, more like mounted infantry than cavalry proper.

The lancer image that drives the image of the medieval knight certainly came out of the Roman cataphract, but it was really the Normans and the Franks who fulfilled this image at the time the Roman empire still existed. Anna Komnene remarks on the devastating impact that Norman and Frankish lancers could have, something that at the time of Alexios Komnenos seemed a distant memory despite the fact the army of Basil II existed only 70 years prior. But the difference was that these men, the crusading Normans and Franks, were typically not armed with composite recurve bows, while the Romans relied heavily on that style of warfare.

For further reading I recommend Michael J. Decker's "The Byzantine Art of War" as an introduction to the topic. Of course Maurice's Strategikon and Leo's Taktika are also worth reading as primary source material (6th and 10th century, respectively).

Also I believe /u/Valkine and /u/Steelcan909 can contribute more on Frankish and Norman warfare.

Late Ancient western cavalry was essentially coming from Late Roman cavalry : Frankish mounted units were already mentioned in the Noticia, but Frankish conquests in Gaul allowed Franks to takeover most of the laetic units stationed there, especially Alans and Taifales. In addition, other Roman cavalry units stationed in Gaul (probably under the rule of hospitalitas, being supported trough land allocation which meant they were tied up to Roman landowners in Gaul, probably as bucellari, a private army) which Procopius mentions one century later.

But as time went on, the Visigoths forced their way into the Roman empire and seized all Spain and the portion of Gaul lying beyond[53] the Rhone River and made them subject and tributary to themselves. By that time it so happened that the Arborychi [Armoricans, Britto-Romans] had become soldiers of the Romans. And the Germans [Franks], wishing to make this people subject to themselves, since their territory adjoined their own and they had changed the government under which they had lived from of old, began to plunder their land and, being eager to make war, marched against them with their whole people. But the Arborychi proved their valour and loyalty to the Romans and shewed themselves brave men in this war, and since the Germans were not able to overcome them by force, they wished to win them over and make the two peoples kin by intermarriage. This suggestion the Arborychi received not at all unwillingly; for both, as it happened, were Christians. And in this way they were united into one people, and came to have great power.

Now other Roman soldiers, also, had been stationed at the frontiers of Gaul to serve as guards. And these soldiers, having no means of returning to Rome, and at the same time being unwilling to yield to their enemy [Goths] who were Arians, gave themselves, together with their military standards and the land which they had long been guarding for the Romans, to the Arborychi and Germans; and they handed down to their offspring all the customs of their fathers, which were thus preserved, and this people has held them in sufficient reverence to guard them even up to my time. For even at the present day they are clearly recognized as belonging to the legions to which they were assigned when they served in ancient times, and they always carry their own standards when they enter battle, and always follow the customs of their fathers. And they preserve the dress of the Romans in every particular, even as regards their shoes.

Basically Franks seems to have largely inherited Late western Roman army and its cavalry; something largely in line with which is archeologically observable.

How did this cavalry fought? As u/FlavivsAetivs said, these cavalrymen knew how to charge pretty well and we even have description of Frankish charges, curtesy of Gregory of Tours.

And Theodoric took his brother Clothar and his son Theodobert to help him and went with his army. And the Thuringi prepared stratagems against the coming of the Franks. For they dug pits in the plain where the fight was to take place, and covering the openings with thick turf they made it seem a level plain. So when they began to fight, many of the Frankish horsemen fell into these pits and it was a great obstacle to them, but when this stratagem was perceived they began to be on their guard. When finally the Thuringi saw that they were being fiercely cut to pieces and when their king Hermenfred had taken to flight, they turned their backs and came to the stream Unstrut. And there such a slaughter of the Thuringi took place that the bed of the stream was filled with heaps of corpses, and the Franks crossed upon them as if on a bridge to the further shore

We learn there that not only Frankish cavalry made charges and were able to do so multiple times in a battle, but that it was probably common enough by the VIth century that their enemies knew which tactics to use to counter them. From what we know of equestrian equipment found in graves, the equipment was similar to Late Roman cavalry and while stirrups might have offered a better balance and a better penetration power,it is not obvious : at least Franks seems to have tought so considering that while fighting with Avars since the VIIth, who used stirrups, they didn't adopted them yet and only did so later. While Avar bodies present some particular stress on their feet, both Frankish and Avar bodies depict what is sometimes called the "horseman syndroma", a series of wounds and marks on the hips.

The relative lack of Frankish cavalry during the Gothic Wars might be explainable to Frankish kings (especially in what became Austrasia) resorting to their German clients or tributary's troops; keeping cavalry as a personal close guard. But most indicate that Franks put horsemanship (in warfare, but as well for hunting and exercising) and horse-breeding in high regard and as a military duty such as described in the Salic Law.

That said, it does seems that cavalry not only meant something quite different from the Byzantine meaning; but eventually from the Late Roman model : cavalry might not have been systematically engaged, or not have played a main tactical role while having similar technics. Contrary to what happened in the East, the infantry took the primary role, although probably as a mounted infantry; only mobilising a large cavalry against largely mounted ennemies (such as Byzantines) and not systematically so : the Battle of Tours, which is sometimes claimed to be the triumph of the Frankish stirrup (which is absent from any historical or archaeological mention) is rather described in the early account as having Franks forming a shield-wall ("like a glacier" says the Chronicle of 754) and unable to pursue their enemy.
It's hard to have a clear idea why, maybe due to a lack of a tactical need besides raiding and counter-raiding (and as well used as an interdiction force such as during the royal faida, to deprive foes from supply), as Frankish infantry/mounted infantry seems to have been mobile and efficient enough in the battlefield.

A good clue would be that, while the lack of stirrups is not a real problem for a lancer or or even a javelineer, it doesn't allow for the use of close combat weapons such as axes or swords; such weapons would remain the prerogative of footmen.

The growth of Carolingian cavalry, by the VIIIth century, would be then due not to Charles Martels' rehabilitation of cavalry (a picture issued from a conception of Frankish armies as a bunch of half-naked savages), but from the incredibly increased mobilising capacities of the new dynasty and of a reduced aristocracy (maybe 60 big families for the whole of Carolingia) who favoured horses as symbol of an aristocratic warfare and status. But even in its Saxon campaigns, cavalry still seems to have mostly a reckon and harassing role, with the occasional charge, especially against infantry as a shock tactic.

Not that light cavalry was unknown in Gaul either, but it tended to be more peripheral : the famed Vascon light cavalry, which might be what allowed Eudon to defeat Arabo-Berbers at Toulouse in 721, and to allow harassment of Arabo-Berbers at Tours in 732, is a perfect example of an elite cavalry in Merovingian Gaul.