I did a quick test in the inspector on both Firefox and Chrome. Setting 'overflow-y' to 'auto' fixes this on Firefox (version 66.0.5 on Windows 8.1). It'll show a scrollbar when the box overflows and removes the scrollbar when there is nothing to scroll.
The experience on Chrome however (version 74.0.3729.131 on Windows 8.1) isn't great regardless what I set it to. When setting it to scroll there are no scrollbars and no other indicators that the box is scrollable except for the text being cut off and the only way to scroll is by using the mouse wheel. Setting 'overflow-y' to auto produces pretty much identical behavior.