Inflight,with a very low OAT ,you can see all pumps illuminated,it happend this year to a few NG ,flying high and with a very low temp.They had to descent to lower FL.It was not a 'major structural failure' as you see ,but can give a ideea of one reason why you can see all hyd lights on.
One other reason I can think off will be a major blast in the wheel well area,but in this case you will feel it for certain in the cockpit
For sure you won't have time ro remark the hyd lights....
The TRE may have read about the DC-10 accident,but there is no resemblance with a 737 (no tail engine to severe all lines,a different pcu for the stby....)
Contaminated hyd liquid may cause the lights to come on,but you'll check during walk-around for this condition.And I doubt would happen to both sys at once.
You can tell him that 'if all pumps lights on-major structural failure ' (big blast maybe) it won't be a problem of continuing or not of the take-off.It would be a problem of emergency evacuation.