The problem is people are thinking of a drill bit and the distance the 'point' travels. For a sphere the size of the earth they are envisaging a drill (of any size but lets say 1mile diameter) and seeing the point touching the north pole and drilling down and exiting the south pole - which is an 8000mile distance.
Think instead of a hole saw (google image hole saw if you dont know what one is - imaging one is essential to the problem) - it isnt a 'solid' drill bit - it is only a perimeter. Now think of a hole saw almost the size of the earth such that when you lower it down over the earth it only cuts through 6 inches of material. You can see that the smaller the hole saw the more material it would cut through - if your hole saw was 1mile wide it would bore through all 8000 miles... if your hole saw is 4000 miles in diamater it cuts through less material. if your hole saw is 8000 miles wide less 6 inches it only bores through a minute amount of material.
This is the core of the problem - people think the length of a hole (that passes exactly through the center of the sphere) is measured from the two most distant points - ie north pole to south pole when it isnt - it is the shortest distance - if you could stand on that huge ring left after hole sawing the earth the 'thickness' as you measure down is 6 inches.