Stirrups, on the whole, did not exist so that people could climb onto horses. I know of only one such example, in early Chinese cavalry, where they had a single ring on a strap on the port side of the saddle. It was used as a mounting step then tucked out of the way. It barely lasted a century.

Horses were domesticated and ridden from c.4500 BCE.

The stirrup was invented c.500 BCE among steppe tribes like the Sauromatians, Sakas, and Massagetae. We see stirrups in sculpture in both the Chertomylek urn and the Kul Olba torc, both created by Hellenic sculptors. Everyone else on the planet ignored it for about a thousand years, because effective stirrups also require a saddle tree to which to attach them.

That's because stirrups exist so you can stand on them while you are on the horse. They allow you to swing large heavy weapons without swinging right off the horse - you put your foot down on that side to retain balance. It let the steppe tribes develop the fully armoured clibanarius or cataphract with a really big sword.

They also let you stand clear of the horse's movement when you use a bow. But all those famous Assyrian and Parthian horse-archers were working without stirrups.

Without stirrups, people got on their horses by vaulting aboard, just like anyone riding bareback today. Did you ever see cowboys in Westerns showing off (Young Guns comes to mind) by mounting without using the stirrups? They swing a leg over or vault on from behind.

It was part of the investiture of a new knight in the High Middle Ages that, in full armour, he swing into the saddle of his destrier without using the stirrups. Being cavalry isn't for weaklings.

Sensible people have always used mounting blocks even when they had stirrups. There were also mounting ladders, which are like library steps.

The Chinese and the Persians were the first outsiders to adopt stirrups for their own cavalry, not just see them on hired steppe mercenaries. (Azzaroli, Early Horsemanship) They were not used in Europe until about the time of Charlemagne. (Osprey's Armies of Charlemagne has a good discussion of that adoption.) Even in the time of the Tudors, the fully armoured Irish gallowglass didn't use stirrups.

The Romans didn't have saddle trees, so no stirrups.

Now, their Gaulish auxiliary cavalry did have saddles. But the design was not derived from the steppes, and they never added stirrups. They relied on a "four-horned" shape that cupped the thighs to stay more secure in action.

So I hope you can see that stirrups are not a necessity in riding, and barely possible without a treed saddle (there is a device like a girth with stirrups called a numnah) They certainly are not required for mounting.