I don't think Aerith would feel close enough to Cloud to hug him like she did with Zack - yes, she was close, but she also knew that it wasn't the real him, which made her cautious, in a way.
That and the compilation says that it's Tifa who supports the mentally weak side of Cloud
I'm not sure that she would have the patience Tifa had during CoT - if anything, Aerith didn't strike me as a patient person during the game XD
But I think that in the end, it's all because of Cloud himself: during the game, he seeks Tifa's approval - even during Disc One, hence the 'only your actions mattered' to Tifa when he was about to become a veggie. I think, because it's Tifa that he loves, he's being cheered up best by her, simply because her words count more than any others'. In AC/C, it was important that he had a talk with Aerith and Zack, because they were the only ones able to free him of the guilt he felt. It's as simple as that - it's not about who can cheer him up the best, it's about whose words he needs.