Not quite that simple - some makes of prop go full fine when the oil pressure is lost, some go full coarse - depends on the governor mechanics. I was going from failing and possibly faulty memory saying the Hamilton Standards on a P-51 goes full coarse . The statement was based on my (admittedly limited) knowledge of the governor mechanism and having watched a number of deadsticks landings over the years at Reno. Will inquire to get a definitive answer from the warbird mechanics who will know for sure. I do know that the Rotol contra rotating props on the Griffon engined Red Baron went full flat when it lost all oil pressure, which slowed the aircraft enough that Steve Hinton was unable to make the runway, resulting in a crash that only by a major miracle he lived through. Pretty sure the P51 HS works the opposite, going full course, so the pilot with a dead engine has a reasonable chance of gliding back to a decent approach.
Also, I believe for props that could be fully feathered (e.g. most WWII era multi-engine aircraft), if the main oil supply was lost there needed to be some form of auxiliary oil pressure source to drive the blades to the full feather position.
cheers
Lo