A major challenge in the burgeoning field of quantum simulation for<br>high-energy physics is the realization of scalable $2+1$D lattice gauge<br>theories on state-of-the-art quantum hardware, which is an essential step<br>towards the overarching goal of probing $3+1$D quantum chromodynamics on a<br>quantum computer. Despite great progress, current experimental implementations<br>of $2+1$D lattice gauge theories are mostly restricted to relatively small<br>system sizes and two-level representations of the gauge and electric fields.<br>Here, we propose a resource-efficient method for quantum simulating $2+1$D<br>spin-$S$ $\mathrm{U}(1)$ quantum link lattice gauge theories with dynamical<br>matter using qudit-based quantum processors. By integrating out the matter<br>fields through Gauss's law, we reformulate the quantum link model in a purely<br>spin picture compatible with qudit encoding across arbitrary spatial<br>dimensions, eliminating the need for ancillary qubits and reducing resource<br>overhead. Focusing first on the spin-$1/2$ case, we construct explicit circuits<br>for the full Hamiltonian and demonstrate through numerical simulations that the<br>first-order Trotterized circuits accurately capture the quench dynamics even in<br>the presence of realistic noise levels. Additionally, we introduce a general<br>method for constructing coupling-term circuits for higher-spin representations<br>$S>1/2$. Compared to conventional qubit encodings, our framework significantly<br>reduces the number of quantum resources and gate count. Our approach<br>significantly enhances scalability and fidelity for probing nonequilibrium<br>phenomena in higher-dimensional lattice gauge theories, and is readily amenable<br>to implementation on state-of-the-art qudit platforms.