Wispy smoke from the dipstick tube is normal immediately after shutdown. There is a hurricane of oil inside when the engine is running. That will create a mist that can look like smoke.
The pump could fail with a punctured diaphragm, but likely would not work well. The crankcase pressure pulses would tend to push oily vapor from the crankcase hurricane through the diaphragm, but that would make the pump weak as it would not develop pulses as strong, and would likely lessen the diaphragm motion, thus weaker or no pumping action. In most designs, a leaky diaphragm would allow fuel into the crankcase... The 'other' side of the diaphragm is exposed to fuel inside the pump chamber where there are two check valves. One draws fuel into the chamber as the spring moves the diaphragm back to its at-rest position. When the pulse occurs, it pushes the diaphragm back and forces fuel out the check valve leading to the float bowl. The pump could leak the pulses if it was not sealed properly, but likely would again not work well. Enough oil on the 'back' side of the diaphragm could hinder diaphram movement, and thus pumping. If oil is leaking from the diaphragm vent hole it could be leftover from tilting/rotating the engine allowing oil to travel from the crankcase.
tom