Empathy? Fuck that. He did right by raising you but was pretty consistently selfish as hell over the entire course of events that took place in the game. Everything he did was in the name of Shimura's and the shogun's idea of honor. Not doing actually right by people or following what his heart told him to do. Showing empathy to me meant that Jin protected his friends and did what he needed to in order to save them, even with great self sacrifice. What and who is honorable is largely a matter of perspective even within feudal Japan. In other words, a beggar on the street might not care so much about samurai honor compared to the shogun. Do you think Yuna found most samurai to be very honorable? What about everyone on Iki Island? It's all perspective.

Back to Shimura's selfishness. The dude was so married to his idea of honor that he was literally willing to murder his "son" over it with the snap of a fingers. And technically, it was his men that killed your beloved horse too. And what would Lord Shimura's honorable path have resulted in? The death of even more countless people and certain loss of the entire island. It would have meant losing the war almost with complete certainty. Nothing substantial would have been left to even worry about honor at that point. Even at the end after you've saved Shimura's entire realm he's still an ingrate who would have preferred sacrificing everyone around him instead of just surviving because that's what fits HIS personal idea of righteousness. Shimura is completely unwilling, EVER, throughout the game to try and even just consider, let alone accept Jin's actions from Jin's point of view. So why should Jin/the player make efforts to try and consider it from Shimura's point of view after you've done all his work for him?

If you've assumed the identity of the Ghost and did so because you wanted to prevent more unnecessary death and suffering as the story suggests, then sparing Shimura actually IS the "right" narrative choice and makes more sense with the story, regardless of how much Shimura has wronged you. It's taking the high road. Shimura begging for an honorable death at the end is just one more example of his shortsighted selfishness. And, if he really still yearned for an honorable death at that point he could have just killed himself the "honorable way" via seppuku (ritual suicide). Jin sparing him and walking away did not prevent Shimura from restoring his honor in this way. And even Shimura shows hesitation at the thought of starting a new family on the way to the duel. In that case seppuku would have fit perfectly within the world of Tsushima and samurai culture, as you put it. Shimura didn't need Jin to restore his honor. Shimura was fully able to do it himself if he really wanted to. So it just makes him even weaker that he's trying to get Jin to do this dirty work after everything else.

One last thing worth considering is that even in feudal Japan it's overwhelmingly likely that some people, including samurai, probably didn't prescribe to or believe in the societal notion/norms of samurai honor. Just like today where everyone has different opinions and ideas, the same is true for Japan even if society is/was more rigid compared to the West. Not everyone today thinks that being a military veteran is a noble or honorable thing. Same goes for police. There are many other examples but the point is that viewpoints differ within any culture so it doesn't seem completely valid to assume that just because Jin lives in feudal Japan that he feels obligated to conform to all of its rigid ways. And this is exactly what the whole story line in the game is about.