The stick by the door isn't uncommon in larger helicopters with some sort of auto hover capability. The problem in a rescue situation is when it's used, you have one man in the back moving the helicopter with one hand, operating the hoist with another, holding the cable with his third hand and securing himself with his fourth. Meanwhile you have two pilots in the cockpit discussing where to have supper.
It works so much better if the pilots fly and the hoist operator hoists.
There are situations where the stick is useful, but not in mountain rescue in general. Overwater rescues with little spatial reference for the pilots is one great area for the hover trim stick.