IIRC, the C-54 was the workhorse of all that.....
Correct!
The C-47 Skytrain played an important part in the lift but the C-54 Skymaster was the cock of the walk. Without the C-54 its doubtful the airlift would have succeeded. With the C-54 they were actually able to deliver more supplies than were strictly required.
That Douglass fella sure knew how to build airplanes.