Using the extra power of the negative adversary method
Apparently, I cannot comment so this will be an answer, even if it only a partial answer.

Element-distinctness has a lower bound of $\Omega(N^{2/3})$ and its certificate complexity is $\sqrt{N}$, so if one tries to prove it using the adversary method, he would need to use the adversary method with negative weights (which is optimal), or why not the Multiplicative adversary method.

Otherwise, the polynomial method is sometimes easier to use that the adversary methods since it suffices to prove the existence of polynomial whereas for the adversary method, you need to explicitly have a good adversary matrix, and compute its operator norm.