The inverter needs a BL-ON [backlight on] signal from the main board to activate it. On the plug where you measured 24.18V, see if one of the pins is marked BL-ON or INV-ON. Should have 2.5 to 3V on it. If not, main board is faulty. Re the 12V going to the main board, are you sure there isn't a STB-3.3V or STB-5V.
Most sets use a permanent Standby 3.3V or 5V to keep the micro going.