That variable has nothing to do with individual signal levels. It works in conjunction with the INTERMEDIATE_POS=1 setting, and causes also the very early iterations of position estimates to be reported to client apps as a "fix".
The unit of ACCURACY_THRESH is meters. Setting it to 3000 means that early position estimates are reported, even if they are still uncertain by 3 km. However, to my knowledge, the variable does not actually influence how the iteration towards the final solution for the position progresses.
To me, the fact that @Name_not_avail saw their phone getting the fix