Uhhh, the turning point in Cap's decision was going to his slampiece's funeral that he never got to slam and her words to live by were "YOLO. Screw everyone else. You do you." And then since he couldn't slam her he slammed her great grandaughter. (grandniece? I forget what she was)
Seems rather emotional to me.
Cap already had his mind made up. He was going to against it whether Carter died or not. So what if it gave him a little motivation.
Iron Man probably would have sided with Cap had a random woman not cornered him and made Tony go all "We can trust the government now because a kid died while we were saving the planet."
Proof? When do we as the audience know that he made up his mind before that? Because if he had already made up his mind before that then the whole eulogy served no purpose whatsoever in the movie. Seems to me you're reaching a bit.
Literally the scene right before he finds out she passed away. Cap is trying to convince them why they are wrong. Pretty sure he even has the last word.