As Tom mentioned, a plug-in tester needs an EGC because its test button simulates a ground fault by connecting a resistor between hot and EGC.
If you have solenoid tester (Wiggy) then connect it between the hot and EGC of a receptacle on the circuit. This should trip the GFCI.
If it doesn't trip, then measure the voltage between the neutral and EGC while the Wiggy or other small load is connected from the hot to EGC (to apply some current to the EGC). If there's a substantial N-EGC voltage then there is a problem with the EGC.
Is this a single pole GFCI circuit? That's likely, but I'm asking just in case it's a 2-pole GFCI breaker with a load neutral for a MWBC. With a MWBC it's possible that there could be some leakage on the other phase that's not enough to trip the GFCI by itself, but which could cancel enough of the leakage applied by the test button to prevent the GFCI from tripping. To do this the leakage on the other phase would have to be around 1mA to 5mA. This is a fairly narrow window and so it's not very likely to happen, but still possible.