That option you have screen shot, will enable timesheet in the new UI. If you enable it, ALL users in the original UI will no longer have access to timesheets in the original UI.
However, your description of the issue experienced is not the above, as some staff have access to Timesheet in the Original UI, and some don't. There has to be something else preventing these staff from accessing Timesheet in the Original UI.
Not providing a solution, and may be stating the obvious.