Designing robotic manipulators for generic tasks while meeting specific requirements is a complex, iterative process involving mechanical design, simulation, control, and testing. New computational design tools are needed to simplify and speed up such processes. This work presents an original formulation of the computational design problem, tailored to help design generic manipulators with strong reachability requirements. The primary challenges addressed in this work are twofold. First, the necessity to consider the design of both continuous quantities and discrete components. Second, the ability to guide the design using high-level requirements, like the robot's workspace, without needing a specific manipulation task, unlike other co-design frameworks. These two challenges are addressed by employing a novel kineto-static formulation, resulting in a Mixed Integer Nonlinear Programming problem, which is solved using bi-level optimization. A compelling use case from a real industrial application is presented to highlight the practical effectiveness of the proposed method.