Given that Rand was told about his birth by Tam that night, even if he didn't know enough to draw his conclusions then, it would've taken precisely one question from Moiraine to the five of them to tell her precisely who the Dragon was - so that she could actually start preparing him, and so that she wouldn't have to risk the lives of the four others by taking them into the Ways. Or alternatively, she might have told them enough for Rand to realize he was the Dragon, but without him being willing to publicly say so - but then as well, by the time they were outside the Waygate in Tar Valon and Moiraine made that speech, surely he'd have spoken up to not unnecessarily risk his friends.

Yes, this is definitely part of the problem - though they have focused on the risk of the Dragon turning to the DO's side, not the risk of the Dragon being a male channeler going insane, and the turning risk could exist also with a female Dragon of course. Though the logic behind their whole plan to go to the Eye of the World is fairly non-existent either way.