Being Paparazzi or not is just a state of mind.
Looking back at the photos later on and in years to come is priceless regardless of the name tagged for what you did.

Just use common sense of course.
If photo taking not allowed, then just follow instruction.
If wife in great pain or has complications, of course do the right thing (whatever is necessary at that time)

17-55 is f2.8 right??
If so, either lens will be fine.
I've photographed with both a 28-75/2.8 zoom and 35/2 prime before.

Hospital for the delivery will have some differences in what you are allowed to take and how good the lighting is.
Either way, with at least f2.8 lens and camera nowadays, its not a problem at all.
Anyway, all the shots will be precious to you

Gd luck and Congrats