The ordered vacancy compounds AM(2)X(4) have gained significant attention in thermoelectric research due to their promising electrical and thermoelectric properties. In this work, we present a first-principles-based high-throughput approach to screen 450 ordered vacancy compounds AM(2)X(4) for potential thermoelectric applications. Initial screening criteria such as band gap, stability, and Pugh's ratio, followed by the phonon spectrum analysis through interception frequency, were used to identify promising candidates, highlighting a strong correlation between lattice thermal conductivity and Pugh's ratio. Despite sharing similar crystal structure of MgM2Te4 (M = Sc, Y, Ga, and In), MgY2Te4 exhibits an exceptionally low lattice thermal conductivity of 0.12 W/mK, in stark contrast to MgGa2Te4 (6.88 W/mK). This discrepancy is attributed to the significantly weaker bonding strength of Sc/Y-Te compared to Ga/In-Te, which enhances lattice anharmonicity and phonon scattering. Additionally, candidate materials with ultralow lattice thermal conductivity are characterized by low-frequency phonons contributing <2 THz, among which MgY2Te4 demonstrates excellent thermoelectric performance with a maximum ZT of similar to 4.7. This work not only unravels the importance of strong anharmonicity and phonon scattering induced by weak-bonded heavy elements on the thermoelectric performance in the AM(2)X(4) family, but also showcases that the screening through Pugh's ratio and interception frequency is an effective strategy for discovering high-performance flexible thermoelectric materials.