Quantum repeaters enable the generation of reliable entanglement across long distances despite the underlying channel noise. Nevertheless, realizing quantum repeaters poses a difficult engineering challenge due to various device constraints and design tradeoffs. Herein, we propose and analyze an efficient hybrid quantum repeater design that integrates atom-based quantum processing units, spontaneous parametric down-conversion photon sources, and atomic frequency comb quantum memories. Our design leverages the strong spectro-temporal multiplexing capability of the quantum memory to enable high-rate elementary-link entanglement generation between repeater nodes. Transferring the photonic entanglement into matter-qubit entanglement, together with deterministic quantum operations, further enables reliable long-distance entanglement distribution. We analyze photon-loss channels in the hybrid architecture and propose suitable error-suppression strategies that are natively incorporated into our repeater protocol. Using numerical simulations, we demonstrate the advantages of our hybrid design for end-to-end secret key rates in a linear repeater-chain model. With continued advances in relevant hardware technologies, we envision that the proposed hybrid design is well-suited for large-scale quantum networks.