To ride people down, neither lances nor stirrups are required. All you need is a horse to, quite literally, ride over people and trample them.

Nor are stirrups required to wield lances. While Roman cavalry in this period did not make heavy use of them, armoured lancers did exist, like for example the cataphracts used in the east (armoured horses + armoured riders) and Alexander's famous companion cavalry. These lances tended to be two handed, unlike the later medieval lance, but were quite capable of being used in a charge.

The whole "stirrup theory" came about when Lynn Townsend White published on the subject in the 60s, but was fairly quickly criticised and attacked at various points, and nowadays nobody really takes it seriously. Stirrups are handy when riding a horse, and they certainly give a nice and stable platform for mounted archery, and allow the rider to stand up in the saddle and wield a sword with more vigour. They also greatly ease mounting up, and we see ancient treatises such as Xenophon's devote considerable attention to the problem of mounting a horse without stirrups. (You can either vault onto the horse, or have a friend or servant give you a leg up.) But they did not revolutionise warfare, and the causes for a perceived cavalry dominance in medieval times are more likely sociological than technological.

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Unfortunately, it's a bit trickier to answer your actual question: how DID Roman cavalry fight in battle?

Roman cavalry armament went through a number of evolutions over the ages, but the time of Caesar is one where we know relatively little about them. We know about the organisation and armament of the professional auxiliary cavalry of Augustus that came after. We know at least a little of the Republican citizen cavalry that came before. But in Caesar's day, the citizen cavalry had disappeared, and cavalry was drawn from allied peoples in a rather ad-hoc fashion. That said, they probably resembled the later professional auxiliary cavalry more than anything, since those were based on practices that had come into being in this period.

The Roman cavalry of Augustus' day were wore mail or scale armour and had heavier helmets than the infantry, carried oval or sometimes rectangular shields, and were armed with long spatha swords (not the short gladius, which as you rightly deduce wasn't practical to use on a horse) and a number of spears or javelins. Some of these spears were long and used for fighting in close quarters, others like the javelins were for throwing. One source specifically mentions cavalrymen carrying one fighting spear and two throwing javelins, though this may not have been the case for all units. Some appear to have carried significantly more javelins.

Either way, Roman cavalry in this period did not use the long, two-handed contus lances I mentioned earlier, nor did they ride armoured horses. Units of Roman cataphracts that did were first raised by Hadrian in the 2nd century, and only became prominent in late antiquity. Roman horses did sometimes wear headgear, though that would be mostly decorative.

Although Romans did not have stirrups, they did have specialised four-horned saddles. (Unlike the Greeks, who had ridden bare-backed for most of their history) Recent experiments with reconstructed Roman saddles indicate that this allowed the rider a very firm seat, as when sitting down the "horns" would press against him from four sides. A rider with such a saddle can charge, throw spears, and even lean quite far from side to side without losing his seat. It wasn't a Roman invention, since Gauls and various eastern peoples also used it, and we're not entirely sure who invented it. Either way, Caesar's Gallic and German cavalry would have used such saddles, and similar armament.

Details on how exactly this cavalry fought is scarce. As your podcast on Caesar has told you, he'll tell us they charge or ride down fleeing troops, but rarely offers more details than that.

The descriptions and the armament we've found suggest that they functioned in a mixed role, engaging both from a distance with javelins and in close quarters with their short lances and long swords. Roman cavalry did not act in the same sledge-hammer like shock fashion that Alexander's companions did, and instead tended to operate on the flanks in the battles we have descriptions of. There, they would engage the enemy cavalry. The winner of the cavalry engagement would then either pursue their fleeing foes, or wheel around to attack the enemy infantry from the flanks or from behind, which could induce them to panic.

In Caesar himself, we frequently read of skirmishes between cavalry and light infantry from both armies, where the legions themselves never even come into play. This type of warfare was also quite important when on campaign, and cavalry was essential to fighting them. Skirmishes like this would not win battles, but they would decide whether an army could forage for supplies, or construct a fortified camp in hostile territory.

The best technical description we have comes not from battle, but from the so-called cavalry games that were used to train cavalry and to let them demonstrate their skill. Arrian describes the practice in the time of the emperor Hadrian: the cavalry would first perform various manoeuvres and charges. Then they would be divided in groups and charge eachother while throwing blunt spears, or charge at stationary targets throwing javelins. Apparently, a well trained cavalryman could throw as many as 15 or even 20 light javelins at a target whilst charging at it. We can well imagine what kind of a hail of missile fire such a cavalry unit could unleash if they carried that many javelins. However, Roman cavalry were certainly also capable of engaging in shock action, and as our other sources show, plenty of units did not carry such large numbers of javelins.

Roman cavalry certainly did not fight as mounted infantry. However, there is one peculiar tactic that Caesar in particular used he would intersperse his German cavalry with fast, lightly armoured German spearmen, who would support the cavalry when fighting enemy cavalry. He mentions the Gauls also usually having light infantry fight alongside cavalry.

The later Roman imperial army did have mixed cohorts of infantry and cavalry, but unlike Caesar's men these do not appear to have fought together.

Sources:

Goldsworthy, Complete Roman Army

Keppie, The Making of the Roman Army: From Republic to Empire