Over the last couple of years, the appearance of the third data release from the Gaia satellite has triggered various wide binary low acceleration gravity tests. Wide binaries with typical total masses ≈1.0 − 1.6M⊙ and separations above a few thousand au probe the low acceleration a ≲ a0 regime, where at galactic and larger scales gravitational anomalies typically attributed to the presence of an as yet undetected dark matter component appear, where a0 ≈ 1.2 × 10−10 m s−2 is the acceleration scale of MOND. Thus, studies of the relative velocities and separations on the plane of the sky, v2D and s2D respectively, of wide binary stars extending to separations above a few kau, provide an independent approach on the empirical study of gravity in the interesting a ≲ a0 acceleration range. Two independent groups, through complementary approaches, have obtained evidence for a departure from Newtonian predictions in the low acceleration regime, in consistency with MOND expectations for wide binary orbits in the Solar Neighbourhood. Two other groups however, have instead reported results showing a clear preference for Newtonian gravity over various MOND alternatives tested, over the same low acceleration regime. We here take a critical look at the various studies in question, from sample selection to statistical treatment of the wide binary relative velocities obtained. We discover a couple of critical problems in the formal design and statistical implementation shared by the two latter groups, and show explicitly how these yield biased conclusions.