We revisit the connection between entanglement entropy and quantum metric in topological lattice systems, and provide an elegant and concise proof of this connection. In gapped two-dimensional lattice models with well-defined tight-binding Hamiltonians, we show that the entanglement entropy is intimately related to the quantum metric of electronic states, and bounded by the Chern number of (Slater-determinant) interacting topological insulators. Our results hold promising applicability to the recently discovered twisted transition metal dichalcogenides, characterized by flat topological bands at low twist angles, where these bounds can be applied at integer filling, and new pathways to enhance entanglement measures by engineering heterostructures with higher Chern numbers.