I've got the staff now and it appears to work as I remember, been jumping up a few walls with it. I recall using that technique to scale a wall to get to the start of laser maze doll run in the Saturn version.
I am relieved from your screenshots to see they put a checkpoint before that. At least I won't have to do the runback after my sure to be many many deaths.
If anything, the checkpoints are very, very generous in this version. Almost too generous.