Chicken Game can be Cosmic Cycloned. Yet it doesn't stop TCG from putting Chicken Game into Forbidden, because the opposing player has to draw Cosmic Cyclone first to get rid of it. And that's if the Cosmic Cyclone isn't getting negated.
There are many ways to counter this card.
Many synchro and other monsters can destroy or banish cards - Rose dragon, Darklord Desire, Treacherous trap hole, Dark lord dragon, Vermillion dragon, Snipe hunter, Alternative dragon, etc. etc.
Armades and Samurai destroer can just attack it without triggering its flipping effect
You can use Forbidden chalice or other effects negating cards
What deck I'm playing is irrelevant. Any deck can have moments where they cannot draw the removal cards quickly. Even you says "not really", indicating that while you draw the removal easily, but there are times when that does NOT happen.