The Federal Aviation Administration introduced the concept of urban air mobility (UAM), a new three-dimensional transport system that operates with a fusion of manned/unmanned aerial vehicles on an urban or intercity scale. The rapid development of UAM has brought innovation and dynamism to many industries, especially in the field of logistics. Various types of unmanned aerial vehicles (UAVs) for use in transport logistics are being designed and produced. UAV logistics refers to the use of UAVs, usually carrying goods and parcels, to achieve route planning, identify risk perception, facilitate parcel delivery, and carry out other functions. This research provides a method for assessing the operational capacity of a UAV logistics route network. The concept of “logistics UAV route network operation capacity” is defined, and a bi-objective optimization model for assessing the route network’s operating capacity is developed. The first objective is to maximize the number of UAV logistics delivery plans that can be executed in a fixed operation time. The second objective is to minimize the total operational impedance value in a fixed operation time. To solve the bi-objective optimization model, the Non-dominated Sorting Genetic Algorithm-II (NSGA-II) is utilized. A UAV logistics route network with 62 nodes is developed to assess the rationale and validity of the proposed concept. The experiments show that with an increase in operation time, the route network’s optimal operational capacity gradually increases, the convergence speed of the algorithm slows down, and the optimization magnitude gradually reduces. Two key parameters—operational safety interval and flight speed—are further analyzed in the experiments. According to the experiments, as the safety interval increases, the route network’s average operational capacity steadily diminishes, as does its sensitivity to the safety interval. The average operational capacity steadily rose with the rise in flight speed, especially when the UAV logistics flight speed was between 10 m/s and 10.5 m/s. In that range, the operational capacity of the route network was substantially impacted by the flight speed.