saying the crash was because the takeoff warning systems wasn't working is not quite right.
Agreed.The TOCW functional check is done during preflight and checked in the BEFORE START checklist.I am guessing the FIRST OFFICER does the initial cockpit set-up.The BEFORE START checklist is then read by the FIRST OFFICER with the CAPTAIN responding.Here is a generic MD80 BEFORE START checklist.
BEFORE START
Preflight Check COMPLETED
FDR Entry SET
Cabin Press SET
Fuel __ Kgs / PUMPS SET
No Smoke ON
Seat Belts ON
Pitot Heat CAPTAIN
Position, Logo & Wing Lights SET
VHF Nav Radios SET
DFGS PILOT FLYING & SET
Fuel Used RESET
Engine Instruments CHECKED
Fire Warning System CHECKED
Gear Door Open Light OFF
GPS SET
Nav / Radio Switch RADIO
Takeoff Warning Throttles CHECKED / IDLE
ADF & Comm Radios SET
Rudder & Aileron Trims ZERO
Takeoff Briefing COMPLETED
--------PAPERS ON BOARD-----------
PMS SET
CG & Flap Wheels ___ / ___, SET
Stabilizer Trim ____, SET
Flap Takeoff Selector ____ SET or STOWED
Zero Fuel Weight _____ Kgs / SET
ART, TRP & Assumed Temperature SET
Speed Bugs & Acceleration Altitude SET
Auxiliary Hydraulic Pump ON for Push back
Two possible scenarios to consider:
A)TOCW functional check performed with aircraft in ground mode(ie prior boarding/loading=no aft CG=weight on wheels).Check is satis.
B)TOCW check omitted.Captain assumes pre-flight items completed and answers "CHECKED,IDLE" based on this assumption.Captain performs subsequent "nuisance" TOCW check(his way of x-checking the functional check) during taxi but aircraft is now in air mode so it remains silent which is what he is looking for.Trap is set.