The first LK-99 paper used this method to claim zero resistivity, but people complained that if the inner probes lost contact, that would also be consistent with their data. This criticism doesn't totally make sense to me, since the apparent superconductivity came and went in the expected way as they changed an external magnetic field. I don't understand how a loose terminal could mimic that figure (I think it was in figure 1).

No, a badly attached probe would usually show a larger resistance, not a smaller one. That's actually the easiest error to make, making improper contact with the sample. The resistance is measured indirectly using a reference current. So you'd measure a higher resistance or a break rather than zero if a probe were not attached correctly (unless the two voltage probes are touching but that would normally speaking be spotted).

The diamagnetism is simply easier to verify using an impure or small sample.