So did you remove the bulb and measure resistance across it or do some other test to verify you have a good bulb? With a bad bulb, you would still see voltage across the bulb. You didn't mention anything about measuring current. Also what is the resistance between the negative side of the dome light and chassis ground? How about between battery positive and the positive of the dome light? (Of course, those measurements are with the bulb out.)
Was the bulb in your photo the bulb you used, or was there any chance you used an LED bulb?
There is one other factor that would allow you to see voltage, but the circuit may not work. That would be corrosion, such as around the door switch. Think of it this way:
Your battery is like a water tower. It's full and ready to supply water. You can verify this by using a pressure gauge (voltmeter). The pipes are all connected, and you open the valve (close the switch). However, you don't see much water coming out (no light). What you didn't realize was that the pipes are old, and a lot of calcium had built up, blocking the pipe (corrosion). If you had a flow meter installed (ammeter), you could measure the low flow. The resistance tests I asked about above are also like testing the pipes, though with the meter, you can get a false low resistance reading.
Good voltage does not equal good current flow.