Let $\mathfrak g$ be a finite simple Lie algebra, and let $r$ denote the ratio of the square length of long roots to that of short roots. Let $\wp>2r$ be an integer and $\zeta$ a primitive $\wp$-th root of unity. Denote by $\mathcal U_\zeta(\widehat{\mathfrak g})$ the Lusztig big quantum affine algebra at root of unity defined by divided powers. In this paper, we establish a current algebra presentation of $\mathcal U_\zeta(\widehat{\mathfrak g})$. Based on this presentation, we construct a $\mathbb Z_\wp$-module quantum vertex algebras $V_{\wp,\tau}^\ell(\mathfrak g)$ for each integer $\ell$. Moreover, we establish a fully faithful functor from the category of smooth weighted $\mathcal U_\zeta(\widehat{\mathfrak g})$-modules of level $\ell$ to the category of $(\mathbb Z_\wp,\chi_\phi)$-equivariant $\phi$-coordinated quasi-modules of $V_{\wp,\tau}^\ell(\mathfrak g)$, where $\chi_\phi:\mathbb Z_\wp\to\mathbb C^\times$ is the group homomorphism defined by $s\mapsto \zeta^s$. We also determine the image of this functor. The structure $V_{\wp,\tau}^\ell(\mathfrak g)$ is substantially different from that of affine vertex algebras. We realize $V_{\wp,\tau}^\ell(\mathfrak g)$ as a deformation of a simpler quantum vertex algebra $V_{\wp,\varepsilon}^\ell(\mathfrak g)$ by using vertex bialgebras,

and decompose $V_{\wp,\varepsilon}^\ell(\mathfrak g)$ into a Heisenberg vertex algebra and a more interesting quantum vertex algebra determined by a quiver.