Thank you for your post!
The Scroll Bar will not operate in Browse Mode when the Field Entry is turned off for Browse Mode. This behavior is expected because before you can scroll any field you must enter it. Something you might do is script a field entry when applicable.
I hope this helps!
This may be just a typo, and if so, know that I'm not trying to nitpick. But, in case this is part of the issue: "Always" and "When Scrolling" are two different options. Do you have "when scrolling" checked? If so, you'd never see the scroll bar since, as TSPigeon points out, you can't scroll a field in browse if you can't enter it in browse mode.
Again, if that's not what you mean, I apologize for seeming nitpicky.