I think I like this idea best...I tried to make a (poor) sketch of it...
The black line is a sliding door rail attached to the wall. The blue circles are the rollers that the stairs "hang" from. There's a similar fixed (non swivel) pair of wheels on the lowest stair that roll parallel to the wall. The sides of the stairs are "open" to allow the hoist arms to go under the stairs. When you want to use the hoist, slide the stairs right so the hoist arms can swing under the car. When you want to use the doorway, the whole thing rolls to the left.
The red safety gate is attached to the green stairs so as you roll the stairs to the right to use the hoist, the safety gate blocks the door to prevent accidents. As you roll it to the left, it just slides along the wall out of the way and opens up the doorway to the now present stairs.
There would be some sort of latch so the stairs don't roll when you don't want them to.