Yes, you're absolutely right that the drag applied by a reel increases as line is removed. This is because the torque required to make the drag slip is constant. So as the lever arm (diameter of the spooled line) decreases due to line being removed, the line tension must increase to apply the same amount of torque that is necessary
to make the drag slip.
However, the size of the reel's arbor has nothing to do with how this changes. In actuality, it is just the width of the spool that makes the difference.