In that brief apartment scene, there was already a spark between Steve and Sharon. Later, after Fury was shot by Winter Soldier, they didn't really have any real scene together except for that encounter at the corridor. When Nat suggested to Steve about calling up Sharon, his reaction was one of interest and not rejection. It's clear that Russo Bros. have a big plan for Steve and Sharon in CW.
As for the board itself, I don't see that many people against this idea aside from you. To claim that "everybody is disgusted with the idea" is hyperbolic and not factually based.
I agree with this. He was definitely into her.