Do all unitary representations weakly converge to zero at infinity?

No, there are simple counterexamples. E.g., take $G = \mathbb{R}$ and $X = \mathbb{C}$ with Lebesgue measure, and define $\rho_t f(z) = f(e^{2\pi i t}z)$ for $t \in \mathbb{R}$ and $f \in L^2(\mathbb{C})$. Then $\rho_t$ is the identity for any integer $t$, so $\rho_n \to {\rm id}$ strongly, not to zero.