There are plenty of situations to get out with it all turning and burning. Fire bucketing is one - a wire tangles on your bucket, or the "sock" doesn't go back; we only carry a light fuel load so a top-up is needed about every 40 minutes - leap out and pump it in by yourself. Shutting down is totally impractical - time wasted in run-down, waiting for TOTs, the count of engine cycles going through the moon.
You are sent to a remote site to collect fire inspectors for a quick recce - get outside the rotor disc, find out what they want, guide them in strap them in, go flying. Land again, get out, get them out, put the seat belts back inside the doors, shut the doors, go flying.
I did hear the story and see the photos of an R22 - the pilot thought there was an unusual vibration, so he landed on the only treeless area around, a rock above a cliff. Not an even surface, and when the pilot got out to examine the problem, he didn't stand on the skid. Maybe the controls weren't frictioned up. Maybe the cg changed, and anyway, the machine started to jiggle. Pilot grabs hold of helo as it jiggles towards edge of rock and tries to pull it back. Then he thinks "What the hell am I doing?", lets go, and watches the machine disappear over the edge.