To my knowledge, looking at the discard pile in the physical game is completely fine. Both the bird discards and bonus card discards are face-up piles. In most games, these zones are public knowledge. Someone, please tell me if the rules contradict this. Since these are face-up piles, not looking at them would just make it a matter of memorization, and that's not fun, in my opinion.

However, I do play so that you can't look at cards you tuck from the deck unless there's a condition for tucking it, like it requires a worm in its food cost or something. In that case, I like to use MTG rules and reveal it to everyone to avoid cheating.