Steve didn't wanna kill the Winter Solider. Yes, he was his mission but his mission was to bring the Winter Solider to justice, to imprison him. Steve decided differently once he realized who that solider was.
"I agree. Cuz she just wants to see his reaction while he sees Peggy. Untouched by age and forgetfulness. His nonreaction and refusal to speak her name break my heart. Cause he know he can't say "Peggy" without breaking down.