It's subtle, but this chapter demonstrated the weakness of horse archers. By being both cavalry and archers they're a bit weaker in both. They typically aren't as heavily armoured as lancers (as this chapter notes) and their horses aren't usually those favoured by either light or heavy cavalry. For horse archers, and steppe people generally, smaller horses with lots of stamina are needed. However, this means that their horses aren't going to be as fast as those of light cavalry, nor able to bear the weight of heavy cavalry.
Also, longbows don't really work well on horseback; they're just too big, and get in the way. Adding to this is that a lack of the best trees for making bows (and fewer trees in general) on either the steppes of Asia in our world, or the mountains in this manga's, limit the materials available for heavy war bows. Composite recurve bows are used out of necessity. Though powerful for their size, these bows lack the power and range of longbows. Note how the enemy archers were able to get in multiple shots before the Mountain People were, and this manga has repeatedly pointed out how the horse archers are all good shots.
In short, horse archers can be amazing units, especially if well trained from childhood, and doubly so if employed against foes not used to fighting them, but they aren't the ultimate unit on any battlefield. I'm glad that this manga doesn't fall prey to the trope that horse archers are the end-all be-all fighting force.