The stairs I had trouble with are the last set of stairs up to the full rooftop. I took the elevator up then you walk some of the roof but to access the last bit it's a long run of stairs up to the final portion with no elevator access. The stairs were not narrow (I'd have preferred that, lol) but they have open stone tracery you can see thru down to the piazza below. My particular fear is heights so this bothered me. Going down I waited until some others were going down and walked with them so I could look at someone's back.
The stairs in the YouTube video are the stairwell going down which I did not find particularly closed in.