The superconducting turbo-electric hybrid propulsion system (TEHPS) integrates superconducting technology and hydrogen energy technology, presenting a potential solution to achieve efficient and high-power propulsion. This study focuses on the design of a liquid hydrogen-cooled superconducting TEHPS, incorporating detailed models for key components, including the hydrogen turbine engine, fuel cell, and superconducting machines. A comprehensive hydrogen-electric-thermal (HET) analysis framework is introduced to optimize system fuel and temperature performance, with feasibility and effectiveness evaluated under conservative, baseline, and optimistic 2035 scenarios. Simulation results for typical mission profiles demonstrate that a hybrid propulsion scheme, combining the engine and fuel cell during takeoff, climb, and cruise phases, and utilizing either the engine or fuel cell alone during the descent phase, can effectively balance fuel and coolant demands, leading to a fuel consumption reduction of up to 22.3% in the optimistic scenario. Improvements in component parameters can significantly reduce the powertrain mass, increase power-to-weight ratio and enhance energy conversion efficiency. Under the optimistic scenario, the system achieves a peak power density of 2.15 kW/kg and an energy conversion efficiency of 75%. Furthermore, a scaled ground testbed for the superconducting TEHPS validated the feasibility of cryogenic cooling, superconducting generators, and hybrid-electric distributed propulsion technologies.