This paper presents the design and development of a novel bioinspired rigid-soft hybrid robotic gripper with versatile grasping primitives and enhanced robustness. It achieves variable joint/tissue stiffness by resorting to hybrid ratchet joints and phalanges. For the first time, the proposed hybrid ratchet joint (HRJ) can selectively switch to four modes to generate tunable flexion/extension locking torque by activating specific clutching ratchets with soft-cell structures. The multi-layer hybrid phalanx can exert suction-lift grasp and adjustable contact stiffness, which are theoretically modeled and experimentally validated by mounting them on articulated fingers. With the proposed HRJs and hybrid phalanges, a tendon-based hybrid gripper is designed and fabricated to evaluate the grasping performance for universal picking and manipulation. Extensive grasping tests are conducted in six representative primitives (including suction-lift and suction-pinch), demonstrating the multi-mode grasping capability in extremely thin and wide objects. Compared with conventional gripper configurations, the proposed hybrid phalanges and HRJs improve the grasping force and enveloping stiffness by 205.1% and 97.9%, respectively.