You are right, but at first propane doesn’t have to draw heat from the outside, as long as liquid’s temperature is well above boiling point. It’s very similar to superheated water:
https://www.youtube.com/watch?v=1_OXM4mr_i0
Of course for such violent effect the depressurization must be very rapid, more like tearing gas cylinder apart than opening the valve. When depressurized at room temperature, the propane is already superheated by about 60 degrees C.
Once the temperature drops, the boiling and evaporation slows down due to required heat transfer, exactly as you said. At this point most of the propane is still in liquid state due to large amounts of heat needed for phase change.
I own hand held 110kW propane torch, and it manages to cool down 9kg gas cylinder in about 2 minutes to the point of limiting it’s output.
