He tries to reel-in Stark a couple of times, when Thor first shows up, and then when he's trying to hack SHIELD. Stark rebuffs him both times.
And fair enough, he should be met with resistance. He should then proceed to prove why he should be in charge.
But no one aside from Stark seems to have any opinion one way or the other about the guy.
And how does he end up being in charge? He just does. And everyone is okay with that.
Stark goes from saying "Following orders isn't my style" and "everything special about you came out of a bottle" to "A living legend who kind of lives up to the legend" and is the one to ask him to call the battle.
What brought that change of heart?
Well, they fixed the Hellicarier together. During which Steve was doing as Tony said. And almost got killed after getting in a fight with three henchmen and needed Tony to save him.
Nothing about that proves Steve lives up to any sort of legend. Certainly doesn't prove that Cap is a leader.
First of all, he is Tony Stark. If there is one personality that was going to clash with Cap's was Tony's. And he tells Tony something that is true, that he's never done the sacrifice play, from which I agree. In previous films, Stark never had to prove himself in the way Captain America, or Thor, or even Hulk/Banner had done in their previous adventures. And in the end he does finally "the sacrifice play". I'll say Cap influenced this too.
And yes, Steve realized that he doesn't know about machines and modern stuff, it was wise to follow Tony's orders. That's also the quality of a leader.