The $\tilde{\alpha} t^{1+1/p} \epsilon^{-1/p}$ bound on gate complexity for simulating time evolution with product formulas is very problem-specific and requires computing nested commutators.
On general grounds, can we say that in terms of the spectral norm $||H||$, the gate complexity cannot scale better than $(||H||t)^{1+1/p}\epsilon^{-1/p}$? The logic is as follows: rescaling the Hamiltonian by an overall prefactor is equivalent to rescaling the evolution time, $e^{-iHt}=e^{-i(H/\alpha)(\alpha t)}$, therefore, a better scaling with Hamiltonian spectral norm would imply that a better scaling with time is achievable.
