The original 1968 design for the Magic Kingdom's Haunted Mansion probably would have used the elevators, it was a "full size" building like the version in Disneyland, and it also could have need the elevators to hide show-building like in DL, but it's position in Magic Kingdom park would have needed the elevators to get guests down 15 feet from "Park Level" to the "Utilidor level"
The original entrance to the 1968 Magic Kingdom Haunted Mansion would have been closer to where the Skyway Station was, or where the Rapunzel bathrooms are today, which is above the Utilidor level... but in 1969/1970 the backstage roadway got re-routed, and Small World which was originally directly behind Cinderella Castle, and behind The Pinocchio Restaurant, got moved West, or "next to" the Pinocchio Restaurant. Which meant the Skyway Station also had to move further West and the Haunted Mansion got scaled down from a Full Sized building like the Disneyland Haunted Mansion, to the "forced perspective" version that was built, at the bottom of the hill, with the entrance "around the back" of the facade building.
The 1971 Version of the building was built "into the ground" and didn't need the elevators
The "exit" crypts in the bottom corner of this next photo are actually a retaining wall the holds back the dirt/earth around the sunken Mansion
The final layout, didn't need the Stretching Room Elevators, and it also didn't really need the Staircase Scene either. The MK Doom Buggy goes down a slope near the Library scene in the beginning of the ride, and come up another slope as you go under Little Leota at the end, which is why she's "in the ride" in Magic Kingdom, they also didn't need the Speed ramps at the end like Disneylan'd Haunted Mansion.
The 1968 model doesn't show the Fantasyland or Cosmic Ray's stages, so it seems possible they recycled them for those locations, but Disneyland had the same type of "rising stages' in late 1960's redo of their Tomorrowland