From the way I read it, you have to invoke the Miracle right when you draw it, while it is still your draw phase before you go to your main phase and before you possibly draw any more cards during your draw phase.
Isn't that how it is supposed to function?
(Since there isn't a reveal clause, I can see how unethical players may try to do a slight of hand and say that another card they have was the one they drew to take advantage of the Miracle cost--how can that be alleviated?)
Yes, Miracle occurs during your draw phase. To prevent massive cheating, make each player set aside the first card drawn on each turn separate from their hand. Similar to activating Delver of Secrets. And if they enter their main phase without activating their Miracle. They forfeit their right to do so.