To add a little to what others have said. The Pseu monitors the landing gear compressed sensors, operates the air/ground relays, sends air/ground discrete signals, provides landing gear position indication and warning, landing gear transfer valve control, landing gear not down warning, speedbrakes extended amber light control, takeoff (aural) warning and door warning.
The cycling of the stab to clear a stab trim disagree nuisance fault does not apply to aircraft built after 12/09 line #3100, or any aircraft that have been updated. One way to eliminate this fault from consideration without doing a bite check is to look at where the stab is positioned. If it isn't at the edge of the green band it's not the source of the fault light.
The scenario the OP described, the fault was a no-dispatch fault. I would have had the pilot perform the bite check. The only potential issue I see would be how it is documented/approved within the airlines maintenance program. Once the fault was known, it could be determined whether it could be a potential nuisance fault or something that needed to be addressed before dispatch.
Lots of information is available if you have access to myboeingfleet.com, such as fleet team digest, maintenance tips, service letters, service bulletins, etc... Just do a search for pseu.