A major challenge in the burgeoning field of quantum simulation forhigh-energy physics is the realization of scalable $2+1$D lattice gaugetheories on state-of-the-art quantum hardware, which is an essential steptowards the overarching goal of probing $3+1$D quantum chromodynamics on aquantum computer. Despite great progress, current experimental implementationsof $2+1$D lattice gauge theories are mostly restricted to relatively smallsystem sizes and two-level representations of the gauge and electric fields.Here, we propose a resource-efficient method for quantum simulating $2+1$Dspin-$S$ $\mathrm{U}(1)$ quantum link lattice gauge theories with dynamicalmatter using qudit-based quantum processors. By integrating out the matterfields through Gauss's law, we reformulate the quantum link model in a purelyspin picture compatible with qudit encoding across arbitrary spatialdimensions, eliminating the need for ancillary qubits and reducing resourceoverhead. Focusing first on the spin-$1/2$ case, we construct explicit circuitsfor the full Hamiltonian and demonstrate through numerical simulations that thefirst-order Trotterized circuits accurately capture the quench dynamics even inthe presence of realistic noise levels. Additionally, we introduce a generalmethod for constructing coupling-term circuits for higher-spin representations$S>1/2$. Compared to conventional qubit encodings, our framework significantlyreduces the number of quantum resources and gate count. Our approachsignificantly enhances scalability and fidelity for probing nonequilibriumphenomena in higher-dimensional lattice gauge theories, and is readily amenableto implementation on state-of-the-art qudit platforms.