In this post, Alfred Centauri describes the resonance as a phenomenon which appears when 'the energy flow from the driving source is unidirectional' and then shows that this is the case for $\Omega=\omega_0$.
However, we know that the resonance is reached for $\Omega = \omega_0 \sqrt{1 - 2 (\beta / \omega_0)^2}$. Knowing that, why isn't: $\omega_0 \sqrt{1 - 2 (\beta / \omega_0)^2}$ the only frequency at which the power is positive $\forall t$ rather than: $\omega_0 $ (for $\beta\neq0$).
edit: Maybe, it's because the frequency corresponding to the maximum energy transfer isn't the same as the one corresponding to the maximum amplitude...
