We study mixed-state entanglement measures in Einstein-Born-Infeld (EBI) massive gravity theory, a model exhibiting both Hawking-Page phase transitions and effective metal-insulator transitions (MIT) at finite temperatures. Our comprehensive investigation reveals that the entanglement wedge cross-section (EWCS), a holographic probe of mixed-state entanglement structure, demonstrates distinctive properties in detecting phase transitions. For the effective MIT, we find that the higher-order terms of EWCS align closely with the crossover temperature, a feature not shared by holographic entanglement entropy (HEE). This sensitivity makes EWCS a complementary geometric tool for probing effective phase transitions at finite temperatures. In Hawking-Page phase transitions, we observe that all entanglement measures effectively diagnose both first-order and second-order phase transitions, with EWCS showing configuration-independent behavior. Importantly, we discover that all geometry-related quantities, including entanglement measures, demonstrate a universal critical exponent of 1/3 near the second-order phase transition point. This result suggests a fundamental connection between quantum information theory and critical phenomena in gravitational systems, and also highlights the potential of EWCS as a powerful probe for phase transitions.