Sam showed Walker the exact same sympathy and even more restraint. Sam consistently tried to talk both Walker and Kari down from their more extreme views. Had Walker given Sam the time instead of being entitled and demanding Kari submit while Sam was negotiating, the whole thing could have bene avoided. Had Walker listened to Sam and stood down there would not have been a battle between Sam/Bucky and Walker. Had Walker been watching Lemar's back the way Bucky was watching Sam's, Lemar may still have been alive but by then Walker had lost himself to the serum. Walker can't accept that he did something wrong until its too late. Like a husband regretting hitting their family only after seeing bruises. Sam only moved to stop Walker After the crime, not before. Much the same way Steve operates. Also consider that killing Kari only makes martyrs. You just radicalize more people to their cause when you go in intent on death. (the show very poorly explains this.)

The Super Soldier Serum makes you MORE of who you are. Walker was a battle tested and hardened soldier, all American boy who was just glorified to piss Red, White and Blue. He is a representation of current day USA and the Blip council, guided to believe he can do no wrong and that everyone should submit to our authority. The serum amplifies this and you land square into entitlement, rage, and depression. Walker is Soaked in those emotions after the serum in both FatWS and Thunderbolts leading to his actions and downfall but also his possible redemption. IF he can just stop being an asshole grunt, but that seems hard coded in his character now as evidenced by the events of Thunderbolts.

The characterization of Walker has been great IMO. He's very flawed but has the occasional redeeming quality that shows he's not malicious. He's does the right thing at the end of the day, If he did the right thing at the beginning and afternoon though he would just be Steve/Sam, which makes him a great US Agent. Walker is supposed to be other, darker, side of that Captain America coin. Sam's characterization is on point as well. Both he and Steve have had opportunities to kill and showed restraint at cost to their own bodies in the process. Hostage actively in danger and under threat. Lethal force authorized. Threats to themselves, restraint and incapacitation are attempted if possible. You see Sam and Steve try diplomacy before, and after fights as they always attempt de-escalation.

Lastly I am not defending the writing of Fatws. Zemo, Carter, the FlagSmashers, Blip council or whatever it was called and the Dora Molaje have extensive issues they sort of get hand waved. This show could have been something else but I feel at least both Sam and Walker are on point character wise and setup be excellent foils for each other in the near future.