Yang Hanying's unacknowledged feelings

How could he allow her to roam freely after his son had failed to eliminate her?

Shadow served as a living, breathing reminder of his son's failure—his own failure.

There was no way he would let her continue to live. 

Yet, when Yang Hanying discovered that he had dispatched his men to pursue her, an overwhelming rage consumed him. In the end, his son issued a chilling ultimatum, threatening to forfeit his position as the successor if Yang Sheng ever laid a hand on Shadow again. Initially dismissing it as mere tantrum, Yang Sheng soon realized it was no empty threat.

Undeterred, Yang Hanying left, independently establishing a successful firearm business. As it thrived, a genuine sense of threat crept over Yang Sheng, prompting him to yield to his son's conditions.

It wasn't the fear of lacking a successor; otherwise, he would have acknowledged Hawk as his son long ago. No, it was the fear of losing his son, Yang Hanying.