The way you suggest should work fine but take care that you don't bugger-up the screwhead underneath as you never know for sure how deep the bung-hole actually is, in some cases.
We've simply used a woodworking scratch awl to gently pry up old bungs poke straight down in the center of the bung. Pry one direction with the grain and then pry side-to-side taking care NOT to damage the perimeter of the bung hole.
After you've done the work needed. Make sure the hole is clean (vacuum) and then seal the new bung in place by dipping the end in a little varnish
(I pour a little in a pie tin wipe excess off by dragging the drips on the side of the plate). LINE
UP wood grains and tap in place with a rubber mallet.
Knock ends off sticking up after its cured a bit with a sharp chisel taking care to keep the angle above the plane of the trim piece.