You have to be able to enter the field before the scroll bar appears so, as you have discovered, denying browse mode access also prevents the scroll bar from working.
Here's a work around that I use:
Define a field of type calculation with text as the result type. If your text field is named TheText, your calculation expression would look like this:
In other words, the name of the original text field is the only thing you put into the specify calculation dialog.
Then put this calculation field on your layout and allow browse mode access. Now you can select the scroll bar, but since this is a calculation field, it still cannot be edited by the user.