Robert E. Howard’s Hyborian Age included some cultures inspired by Asian cultures, but unarmed fighting isn’t really specific to Asian sultures. Rather, unarmed styles evolved in cultures where people weren’t allowed to carry weapons (similar to weapons-that-are-not-weapons, such as various farming tools and laundry poles). In cultures where it was allowed (or in some cases, required) to carry weapons in public, unarmed fighting styles were developed to support the armed martial arts. Medieval European swordfighting treatises, for example, include instructions for kicking, wrestling, etc.
From what we’ve seen, most cultures in the Hyborian Age seemed to allow carrying swords etc. in public, so there was little need to develop dedicated unarmed martial arts. Admittedly, we don’t know a lot about Khitai, since Howard never described it in detail. Of course, this was also a time before wuxia became popular in the West.
Being unarmed is, and should be, always at a disadvantage against armed opponents. Sure, punching and kicking should always be an option, but using a sword, or a club, or a pointy stick, should be more effective.
From a gameplay point of view: It would make a lot of aspects in a survival combat game redundant if you could just develop an unarmed combat style and be reasonably effective with it. So rather than harvesting rare resources to make the best weapons, just punch that giant spider in the face.
Unarmed combat would be cool in some other fantasy subgenre, but for Howardian barbarian fantasy, it’s the axe that makes you a king.