You can narrow it down faster by using a flat blade screwdriver and temporary shorting the C/E pins of the 100 ohms transistors, one by one when the display shows zeros, the faulty segment should light up,
if it does the fault is from the transistor to the chip,
if it doesn't its from the transistor to the display.
if it does the fault is from the transistor to the chip,
if it doesn't its from the transistor to the display.