Had the same problem about a year ago. Was erratic for a while then one day stopped working completely. Went through the same steps as you- had continuity at the connector in your pic BUT had almost no voltage at the reverse bulb sockets.
Turned out to be that the terminations at the reverse switch were so heavily corroded I was dropping almost all the voltage there due to the high resistence. I just cleaned them up and the lights have been working consistently ever since. The revese switch can be reached from under the car, no need to remove it.