You "can't" get a plug stuck in a fly cutter.

A hole saw is a tubular piece of steel, one end ground and set into saw teeth, the opposite end enclosed with a (tapped) through hole for an arbor.

The reason the plug gets stuck in a hole saw is that the tooth set toward the ID of the saw makes a very nice grip on the edge of the plug.

The method you suggest helps, because when you cut through, most of the plug will be proud of the saw, leaving space to grip it and pull it off the centering bit. It also helps breakout on the exit side of the hole.

Another thing you could try are a newer hole saw configuration that is part fly cutter and part hole saw. The cylinder of the hole saw is still there, but there is a large diagonal cut up the side of the cylinder to clear chips. At the base of the cut is a single piece of carbide on the leading edge that does the actual cutting. They cut very, very fast and they don't hold the plug like a "regular" hole saw does.

• http://www.geocities.com/budb3/arts/toolp/flycut.html