Welcome smig!!
I think it depends on the type of tprop.
Free-power turbines, such as the PT-6 on the King Air and the GE on the SAAB, don't require that the prop turns in order to start the engine. So it's easier to let them bleed off to feather, or some procedures (I'm reaching way back) and systems are designed to place the prop to feather prior to shutdown. Usually bringing the condition levers to cutoff will feather the engine as it shuts down (I think this is the way it was in the King Airs - again, I'm reaching deep in the memory, so I could be completely off base)
Direct-drive turboprops, such as the Garrett TPE331 series found on J31s, MU2s and Aerocommanders and the Rolls-Royce Darts found on the old F27 whistle pigs and the YS-11s, require the blades to be in fine pitch during start. Otherwise, the drag on the blades would be too great for the starter to get the engine going. The physics is the same, so if the oil pressure drops, the prop wants to feather. There are usually pins that are inserted into the prop hub, that as the engine spins down, the centrifugal force that holds them away, reduces allowing the pins to fall into position. Forcing the prop to stay in the flat pitch can be done by several procedures. If you hear a TPE331 start up, after the engines get spooled, you'll hear a quick burst of reverse pitch. That backs the hub off the pins allowing them to retract, allowing the prop to coarsen the pitch.