The start button lamp is part of the lamp matrix. If you are on free play it should flash. If on coin, only when there are credits will it flash.

So given you have a good lamp, socket and wiring the problem may in the lamp matrix.

So you should go into lamp test and check your lamps associated with the row and column associated with the start button lamp.

Look at the lamp matrix chart on manual page 3.2 -- manual available on ipdb.

If all the lamps test good (in the row and column) associated with the start button lamp, then you problem is not board related.

All of the controlled lamps have a diode in their circuit to the driver board. You can make some fundamental continuity tests between the driver board and your non-working controlled lamp.

The Yellow/Blue wire on the lamp socket is wired directly to J137 pin 9 on the driver board. This is power in to the lamp.
So you should have continuity between the Yellow/Blue lamp connector to J137 pin 9 on the driver board.

The other wire on the lamp socket goes to the coin door interface board and is connected at J6 pin 4. This is connected to the anode of diode D1.

The arrow/line side or cathode is routed to J1 pin 2. You should have continuity from the cathode side of the diode to the driver board J133 pin 9 (Red/Gray wire.) This is connected to a circuit that provides ground when the light is pulsed.

You can test the diode on the coin door interface board using diode test for D1 as has been previously suggested.

In the remote case there are two diodes on your push button switch, one diode for the switch and another for the lamp it may be oriented incorrectly. You can just swap the lamp connectors to the socket.