Back Gears: If you study the Headstock page of any 10F illustrated parts list dated from 1946 up to the final one in 1966, they are all the same as there were no changes to that part of the headstock. The back gears are engaged and disengaged through the actions of two eccentrics, 10-251 and 10-252. For the back gears to remain disengaged or engaged when subjected to vibration and when engaged to torque reaction while driving the spindle, the two 10-167 Wave Washers must be present and slightly compressed and the eccentrics must be on one side of their peak when engaged and the other side when disengaged. If the eccentrics don't cross over their peaks while being engaged, torque forces and vibration will combine to make the lever slowly move back to the disengage position.
With the motor off, watch the motion of the back gears as you move the lever to engage them. They should move toward the spindle and then back away from the spindle slightly, indicating that the eccentrics are rotating past their peak when going from disengaged to engaged. If this is the case, all that I can suggest it to loosen the set screws in either or both collars and move them slightly on the shaft in the direction to squeeze the wave washers a little more. If the gears move toward the spindle but bottom out and stop before breaking over center, you need to increase the thickness of the 10-261shim pack. If the 10-254 Handle hits the headstock before the eccentrics cross over their peaks, the only solution that I see is to make sure that the handle is tight, mark the headstock side of it, remove and and bend the handle away from the headstock.
As an aside, given that the lathe has obviously been repainted, it would not surprise me any to find that there are no 10-261 shims present.