as a foot archers can use their whole body to pull the bowstring and horse archers only their arms and shoulder

Huh? Sorry, former competitive archer here. I'm trying to figure when I ever used my feet or leg muscles to draw the bow. Not even my buttocks. We all draw with arms, shoulders, and back, and those muscles are available when sitting. Riders don't sit with their knees up at hip level: the thighs are more vertical than horizontal.

EDIT

But back to OP.

Crossbows, as ever, are a wonderful weapon to shoot from behind the crenellations of a wall. The Chinese even mounted big ones on stands to be used as short-range artillery. For a walled city or from on top of a border wall, they get range from height above their opponents, and the slow rate of fire doesn't much matter.

Now the Xiongnu were not Mongols, nor Huns, but had cultural similarities. At later periods they were pretty Han in acquired habits.

Both cultures had a long tradition of professional bowyers making powerful recurved composite bows, that had to be stored in bowcases that preserved the shape. But the crossbow, as in Europe, was easier for the new soldier to use. It could be drawn and held by the mechanism, and its stock was the ancestor of the rifle stock. So, for a conscript peasant army, this "rifle" that fired a bolt was the best and easiest way to give them weapons parity of a kind with the Xiongnu.

Not that an individual crossbowman could equal a horse archer in a field battle. But when you have a thousand or ten thousand crossbowmen, firing in volleys, the speed difference kind of disappears.

In a field battle, you can break down that the horse archer has height from being in the saddle, a much faster rate of fire, and mobility, even if the ballistics of the weapons were equal.

For use of the bow, you need to remember that they were in saddles. Saddles were inve ted by the western steppe tribes for warfare, which eventually were passed with the horse to the easternpeoples beyond Dzungaria. While Assyrian horse archers were impressive (Spruytte did some great re-creations of their techniques), they were riding bareback on saddle cloths. A saddle gives one greater security in twisting around. It also attaches the stirrups, in this case the round "shelf" stirrup, rather than a narrow bar. This lets you put your feet down solidly and stand in the saddle, so that your legs can cancel out most of the horse motion and you can shoot over the horse's head.

So the horse added height and mobility without reducing shooting capability at all.

So while the Chinese may have recommended the crossbow to deter Xiongnu attacks, I wouldn't say that was a claim that crossbowmen were "better" than horse archers, just the Chinese attempt to counter them.